Y. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Stud. Appl. Math. 65, 113–158 (1981).

[Crossref]

M. J. Ablowitz and J. F. Ladik, “A difference scheme and inverse scattering,” Stud. Appl. Math. 55, 213–229 (1976).

[Crossref]

M. J. Ablowitz and J. F. Ladik, “Nonlinear differential-difference equations,” J. Math. Phys. 16, 598–603 (1975).

[Crossref]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, 1981).

N. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

[Crossref]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010).

E. Agrell, A. Alvarado, and F. R. Kschishang, “Implications of information theory in optical fibre communications,” Philos. Trans. R. Soc. A 374, 20140438 (2016).

[Crossref]

F. Ahmad and M. Razzagh, “A numerical solution to the Gel’fand-Levitan-Marchenko equation,” Appl. Math. Comput. 89, 31–39 (1998).

P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).

[Crossref]

E. Agrell, A. Alvarado, and F. R. Kschishang, “Implications of information theory in optical fibre communications,” Philos. Trans. R. Soc. A 374, 20140438 (2016).

[Crossref]

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

N. A. Shevchenko, J. E. Prilepsky, S. A. Derevyanko, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the per soliton capacity of the nonlinear optical fibre channel,” in IEEE Information Theory Workshop (ITW), Jeju Island, Korea, 2015, pp. 104–108.

S. T. Le, J. E. Prilepsky, P. Rosa, J. D. Ania-Castanon, and S. K. Turitsyn, “Nonlinear inverse synthesis for optical links with distributed Raman amplification,” J. Lightwave Technol. 34, 1778–1786 (2016).

[Crossref]

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101, 123903 (2008).

[Crossref]

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).

[Crossref]

J. D. Ania-Castañón, “Quasi-lossless transmission using second-order Raman amplification and fibre Bragg gratings,” Opt. Express 12, 4372–4377 (2004).

[Crossref]

A. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in European Conference on Optical Communication (ECOC), Germany, 2016, paper Th.3.B.2.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.2.

H. Buelow, V. Aref, and W. Idler, “Transmission of waveform determined by 7 eigenvalues with PSK-modulated spectral amplitude,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.E.2.

A. Arico, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo 48, 75–88 (2011).

[Crossref]

J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, “Fast and backward stable computation of roots of polynomials,” SIAM J. Matrix Anal. Appl. 36, 942–973 (2015).

[Crossref]

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 13–16.

N. A. Shevchenko, J. E. Prilepsky, S. A. Derevyanko, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the per soliton capacity of the nonlinear optical fibre channel,” in IEEE Information Theory Workshop (ITW), Jeju Island, Korea, 2015, pp. 104–108.

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).

[Crossref]

R. I. Killey and C. Behrens, “Shannon’s theory in nonlinear systems,” J. Mod. Opt. 58, 1–10 (2011).

[Crossref]

L. L. Frumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method for solving the direct Zakharov-Shabat scattering problem,” J. Opt. Soc. Am. B 32, 290–295 (2015).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method of the fiber Bragg grating synthesis,” J. Opt. Soc. Am. B 24, 1451–1457 (2007).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, “Finite Bragg grating synthesis by numerical solution of Hermitian Gel’fand-Levitan-Marchenko equations,” J. Opt. Soc. Am. B 23, 2040–2045 (2006).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett. 113, 013901 (2014).

[Crossref]

B. Deconinck, H. Heil, A. Bobenko, M. Van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Math. Comput. 73, 1417–1443 (2004).

[Crossref]

G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Comput. Phys. 102, 252–264 (1992).

[Crossref]

P. Boito, Y. Eidelman, L. Gemignami, and I. Gohberg, “Implicit QR with compression,” Indag. Math. 23, 733–761 (2012).

[Crossref]

H. Buelow, V. Aref, and W. Idler, “Transmission of waveform determined by 7 eigenvalues with PSK-modulated spectral amplitude,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.E.2.

A. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in European Conference on Optical Communication (ECOC), Germany, 2016, paper Th.3.B.2.

H. Bülow, “Experimental demonstration of optical signal detection using nonlinear Fourier transform,” J. Lightwave Technol. 33, 1433–1439 (2015).

[Crossref]

H. Bülow, “Experimental assessment of nonlinear Fourier transformation based detection under fiber nonlinearity,” in European Conference on Optical Communications (ECOC), Cannes, France, 2014, paper We.2.3.2.

H. Bülow, “Nonlinear Fourier transformation based coherent detection scheme for discrete spectrum,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2015, paper W3K.2.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.2.

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147, 166–186 (1998).

[Crossref]

S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).

[Crossref]

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147, 166–186 (1998).

[Crossref]

Q. Zhang and T. H. Chan, “A spectral domain noise model for optical fibre channels,” in IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, 2015, paper We-AM2-9.1.

Q. Zhang and T. H. Chan, “A Gaussian noise model of spectral amplitudes in soliton communication systems,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 455–459.

Q. Zhang and T. H. Chan, “Achievable rates of soliton communication systems,” in IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, 2016, pp. 605–609.

A. R. C. X. Liu, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7, 560–568 (2013).

[Crossref]

C. Xi, L. Xiang, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Technical Digest of Optical Fiber Communication Conference and Exposition (OFC/NFOEC) and the National Fiber Optic Engineers Conference (2012), paper OTh3C.1.

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).

[Crossref]

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 13–16.

M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication Systems and Networks (Artech House, 2013).

S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).

[Crossref]

C. Swierczewski and B. Deconinck, “Computing Riemann theta functions in Sage with applications,” Math. Comput. Simul. 127, 263–272 (2016).

[Crossref]

B. Deconinck, H. Heil, A. Bobenko, M. Van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Math. Comput. 73, 1417–1443 (2004).

[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett. 113, 013901 (2014).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: linearization of the lossless fiber channel,” Opt. Express 21, 24344–24367 (2013).

[Crossref]

S. A. Derevyanko, “Appearance of bound states in random potentials with applications to soliton theory,” Phys. Rev. E 84, 016601 (2011).

[Crossref]

S. K. Turitsyn and S. A. Derevyanko, “Soliton-based discriminator of noncoherent optical pulses,” Phys. Rev. A 78, 063819 (2008).

[Crossref]

S. A. Derevyanko and J. E. Prilepsky, “Random input problem for the nonlinear Schrödinger equation,” Phys. Rev. E 78, 046610 (2008).

[Crossref]

S. A. Derevyanko, “Design of a flat-top fiber Bragg filter via quasi-random modulation of the refractive index,” Opt. Lett. 33, 2404–2406 (2008).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasi-lossless and lossy optical fiber spans,” J. Opt. Soc. Am. B 24, 1254–1261 (2007).

[Crossref]

J. E. Prilepsky and S. A. Derevyanko, “Breakup of a multisoliton state of the linearly damped nonlinear Schrödinger equation,” Phys. Rev. E 75, 036616 (2007).

[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and D. A. Yakushev, “Statistics of a noise-driven Manakov soliton,” J. Phys. A 39, 1297–1309 (2006).

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Fokker-Planck equation approach to the description of soliton statistics in optical fiber transmission systems,” J. Opt. Soc. Am. B 22, 743–752 (2005).

[Crossref]

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Non-Gaussian statistics of an optical soliton in the presence of amplified spontaneous emission,” Opt. Lett. 28, 2097–2099 (2003).

[Crossref]

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

N. A. Shevchenko, J. E. Prilepsky, S. A. Derevyanko, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the per soliton capacity of the nonlinear optical fibre channel,” in IEEE Information Theory Workshop (ITW), Jeju Island, Korea, 2015, pp. 104–108.

M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication Systems and Networks (Artech House, 2013).

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper Tu2A.1.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lawery, “Digital fiber nonlinearity compensation: toward 1-Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).

[Crossref]

P. Boito, Y. Eidelman, L. Gemignami, and I. Gohberg, “Implicit QR with compression,” Indag. Math. 23, 733–761 (2012).

[Crossref]

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).

[Crossref]

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lawery, “Digital fiber nonlinearity compensation: toward 1-Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).

[Crossref]

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the nonlinear Shannon limit,” J. Lightwave Technol. 28, 423–433 (2010).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.B.2.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper Tu2A.1.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).

[Crossref]

R. J. Essiambre, R. W. Tkach, and R. Ryf, “Fiber nonlinearity and capacity: single mode and multimode fibers,” in Optical Fiber Telecommunications, I. Kaminow, T. Li, and A. E. Willner, eds., 6th ed., Vol. VIB of Systems and Networks (Academic, 2013), Chap. 1, pp. 1–43.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010).

[Crossref]

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).

[Crossref]

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

E. Meron, M. Feder, and M. Shtaif, “On the achievable communication rates of generalized soliton transmission systems,” arXiv:1207.0297 (2012).

L. Fermo, C. van der Mee, and S. Seatzu, “Numerical solution of the direct scattering problem for the nonlinear Schrödinger equation,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 5–8.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).

[Crossref]

E. Forestieri and M. Secondini, “The nonlinear fiber-optic channel: modeling and achievable information rate,” in Progress in Electromagnetics Research Symposium (PIERS), Prague, Czech Republic, 2015, pp. 1276–1283.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010).

[Crossref]

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).

[Crossref]

J. Frauendiener and C. Klein, “Computational approach to hyperelliptic Riemann surfaces,” Lett. Math. Phys. 105, 379–400 (2015).

[Crossref]

L. L. Frumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method for solving the direct Zakharov-Shabat scattering problem,” J. Opt. Soc. Am. B 32, 290–295 (2015).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method of the fiber Bragg grating synthesis,” J. Opt. Soc. Am. B 24, 1451–1457 (2007).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, “Finite Bragg grating synthesis by numerical solution of Hermitian Gel’fand-Levitan-Marchenko equations,” J. Opt. Soc. Am. B 23, 2040–2045 (2006).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett. 113, 013901 (2014).

[Crossref]

P. Boito, Y. Eidelman, L. Gemignami, and I. Gohberg, “Implicit QR with compression,” Indag. Math. 23, 733–761 (2012).

[Crossref]

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

P. Boito, Y. Eidelman, L. Gemignami, and I. Gohberg, “Implicit QR with compression,” Indag. Math. 23, 733–761 (2012).

[Crossref]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).

[Crossref]

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).

[Crossref]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).

[Crossref]

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic, 2006).

I. T. Lima, V. S. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Computational complexity of nonlinear transforms applied to optical communications systems with normal dispersion fibers,” IEEE Photonics Conference (IPC), Reston, VA, 2015, pp. 277–278.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

S. Hari, M. I. Yousefi, and F. R. Kschischang, “Multieigenvalue communication,” J. Lightwave Technol. 34, 3110–3117 (2016).

[Crossref]

S. Hari and F. R. Kschischang, “Bi-directional algorithm for computing discrete spectral amplitudes in the NFT,” J. Lightwave Technol. 34, 3529–3537 (2016).

[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

S. Hari, F. Kschischang, and M. Yousefi, “Multi-eigenvalue communication via the nonlinear Fourier transform,” in 27th Biennial Symposium on Communications (QBSC), Ontario, Canada, 2014, pp. 92–95.

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).

[Crossref]

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101, 123903 (2008).

[Crossref]

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper Tu2A.1.

M. Tan, P. Rosa, I. D. Phillips, and P. Harper, “Long-haul transmission performance evaluation of ultra-long Raman fiber laser based amplification influenced by second order co-pumping,” in Asia Communications and Photonics Conference, OSA Technical Digest (Optical Society of America, 2014), paper ATh1E.4.

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.B.2.

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11, 395–399 (1993).

[Crossref]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, 1995).

B. Deconinck, H. Heil, A. Bobenko, M. Van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Math. Comput. 73, 1417–1443 (2004).

[Crossref]

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).

[Crossref]

E. Iannoe, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).

[Crossref]

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.2.

H. Buelow, V. Aref, and W. Idler, “Transmission of waveform determined by 7 eigenvalues with PSK-modulated spectral amplitude,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.E.2.

A. Maruta, A. Toyota, Y. Matsuda, and Y. Ikeda, “Experimental demonstration of long haul transmission of eigenvalue modulated signals,” Tyrrhenian International Workshop on Digital Comminications (TIWDC), Florence, Italy, 2015, pp. 28–30.

V. Kotlyarov and A. Its, “Explicit formulas for the solutions of a nonlinear Schrodinger equation,” Doklady Akad. Nauk Ukrainian SSR Ser. A 10, 965–968 (1976), http://arxiv.org/abs/1401.4445v1 .

P. Frangos and D. Jaggard, “A numerical solution to the Zakharov-Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).

[Crossref]

S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part I: theory and numerical methods,” Opt. Express 24, 18353–18369 (2016).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part II: eigenvalue communication,” Opt. Express 24, 18370–18381 (2016).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.B.2.

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform based transmissions with high order QAM formats,” in European Conference on Optical Communications (ECOC), Germany, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Optical communication based on the periodic nonlinear Fourier transform signal processing,” in IEEE 6th International Conference on Photonics (ICP), Kuching, Malaysia, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform based optical communication systems in a band-limited regime,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), Vancouver, Canada, 2016, paper JTu4A.34.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101, 123903 (2008).

[Crossref]

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London A 361, 413–446 (1978).

D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM J. Appl. Math. 31, 121–133 (1976).

[Crossref]

N. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

[Crossref]

P. Kazakopoulos and A. L. Moustakas, “On the soliton spectral efficiency in nonlinear optical fibers,” in IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, 2016, pp. 610–614.

S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).

[Crossref]

P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).

[Crossref]

R. I. Killey and C. Behrens, “Shannon’s theory in nonlinear systems,” J. Mod. Opt. 58, 1–10 (2011).

[Crossref]

S. Oda, A. Maruta, and K. Kitayama, “All-optical quantization scheme based on fiber nonlinearity,” IEEE Photon. Technol. Lett. 16, 587–589 (2004).

[Crossref]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

J. Frauendiener and C. Klein, “Computational approach to hyperelliptic Riemann surfaces,” Lett. Math. Phys. 105, 379–400 (2015).

[Crossref]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, 1995).

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).

V. Kotlyarov and A. Its, “Explicit formulas for the solutions of a nonlinear Schrodinger equation,” Doklady Akad. Nauk Ukrainian SSR Ser. A 10, 965–968 (1976), http://arxiv.org/abs/1401.4445v1 .

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010).

[Crossref]

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).

[Crossref]

S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).

[Crossref]

S. Hari, F. Kschischang, and M. Yousefi, “Multi-eigenvalue communication via the nonlinear Fourier transform,” in 27th Biennial Symposium on Communications (QBSC), Ontario, Canada, 2014, pp. 92–95.

S. Hari, M. I. Yousefi, and F. R. Kschischang, “Multieigenvalue communication,” J. Lightwave Technol. 34, 3110–3117 (2016).

[Crossref]

S. Hari and F. R. Kschischang, “Bi-directional algorithm for computing discrete spectral amplitudes in the NFT,” J. Lightwave Technol. 34, 3529–3537 (2016).

[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: numerical methods,” IEEE Trans. Inf. Theory 60, 4329–4345 (2014).

[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: spectrum modulation,” IEEE Trans. Inf. Theory 60, 4346–4369 (2014).

[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: mathematical tools,” IEEE Trans. Inf. Theory 60, 4312–4328 (2014).

[Crossref]

E. Agrell, A. Alvarado, and F. R. Kschishang, “Implications of information theory in optical fibre communications,” Philos. Trans. R. Soc. A 374, 20140438 (2016).

[Crossref]

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lawery, “Digital fiber nonlinearity compensation: toward 1-Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).

[Crossref]

M. J. Ablowitz and J. F. Ladik, “A difference scheme and inverse scattering,” Stud. Appl. Math. 55, 213–229 (1976).

[Crossref]

M. J. Ablowitz and J. F. Ladik, “Nonlinear differential-difference equations,” J. Math. Phys. 16, 598–603 (1975).

[Crossref]

G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).

[Crossref]

S. T. Le, S. Wahls, D. Lavery, J. E. Prilepsky, and S. K. Turitsyn, “Reduced complexity nonlinear inverse synthesis for nonlinearity compensation in optical fiber links,” in Proceedings of Conference on Lasers and Electro-Optics/Europe and the European Quantum Electronics Conference (CLEO/Europe-EQEC), Munich, Germany, 2015, paper CI_3_2.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lawery, “Digital fiber nonlinearity compensation: toward 1-Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).

[Crossref]

S. T. Le, J. E. Prilepsky, P. Rosa, J. D. Ania-Castanon, and S. K. Turitsyn, “Nonlinear inverse synthesis for optical links with distributed Raman amplification,” J. Lightwave Technol. 34, 1778–1786 (2016).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part II: eigenvalue communication,” Opt. Express 24, 18370–18381 (2016).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part I: theory and numerical methods,” Opt. Express 24, 18353–18369 (2016).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).

[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).

[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22, 26720–26741 (2014).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.B.2.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, and S. K. Turitsyn, “Digital backpropagation in the nonlinear Fourier domain,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 445–449.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper Tu2A.1.

A. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in European Conference on Optical Communication (ECOC), Germany, 2016, paper Th.3.B.2.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform based transmissions with high order QAM formats,” in European Conference on Optical Communications (ECOC), Germany, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Optical communication based on the periodic nonlinear Fourier transform signal processing,” in IEEE 6th International Conference on Photonics (ICP), Kuching, Malaysia, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform based optical communication systems in a band-limited regime,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), Vancouver, Canada, 2016, paper JTu4A.34.

S. T. Le, S. Wahls, D. Lavery, J. E. Prilepsky, and S. K. Turitsyn, “Reduced complexity nonlinear inverse synthesis for nonlinearity compensation in optical fiber links,” in Proceedings of Conference on Lasers and Electro-Optics/Europe and the European Quantum Electronics Conference (CLEO/Europe-EQEC), Munich, Germany, 2015, paper CI_3_2.

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).

[Crossref]

I. T. Lima, V. S. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Computational complexity of nonlinear transforms applied to optical communications systems with normal dispersion fibers,” IEEE Photonics Conference (IPC), Reston, VA, 2015, pp. 277–278.

A. R. C. X. Liu, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7, 560–568 (2013).

[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

Y. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Stud. Appl. Math. 65, 113–158 (1981).

[Crossref]

J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, “Fast and backward stable computation of roots of polynomials,” SIAM J. Matrix Anal. Appl. 36, 942–973 (2015).

[Crossref]

P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).

[Crossref]

S. V. Manakov, “On the theory of two-dimensional stationary self focussing of electromagnetic waves,” Sov. Phys. J. Exp. Theor. Phys. 38, 248–253 (1974).

S. Oda, A. Maruta, and K. Kitayama, “All-optical quantization scheme based on fiber nonlinearity,” IEEE Photon. Technol. Lett. 16, 587–589 (2004).

[Crossref]

A. Toyota and A. Maruta, “Wavelength division multiplexed optical eigenvalue modulated system,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 43–45.

A. Maruta, A. Toyota, Y. Matsuda, and Y. Ikeda, “Experimental demonstration of long haul transmission of eigenvalue modulated signals,” Tyrrhenian International Workshop on Digital Comminications (TIWDC), Florence, Italy, 2015, pp. 28–30.

A. Maruta and Y. Matsuda, “Polarization division multiplexed optical eigenvalue modulation,” in International Conference on Photonics in Switching (PS), Florence, Italy, 2015, pp. 265–267.

H. Terauchi, Y. Matsuda, A. Toyota, and A. Maruta, “Noise tolerance of eigenvalue modulated optical transmission system based on digital coherent technology,” in 19th OptoElectronics and Communications Conference/Australian Conference on Optical Fibre Technology (OECC/ACOF), Melbourne, Australia, 2014, p. 542.

Y. Matsuda, H. Terauchi, and A. Maruta, “Design of eigenvalue-multiplexed multi-level modulation optical transmission system,” in 19th OptoElectronics and Communications Conference/Australian Conference on Optical Fibre Technology (OECC/ACOF), Melbourne, Australia, 2014, paper TH12B3.

A. Maruta, “Eigenvalue modulated optical transmission system (invited),” in 20th OptoElectronics and Communications Conference (OECC), Shanghai, China, 2015, paper JThA.21.

H. Terauchi and A. Maruta, “Eigenvalue modulated optical transmission system based on digital coherent technology,” in 10th Conference on Lasers and Electro-Optics Pacific Rim, and the 18th OptoElectronics and Communications Conference/Photonics in Switching (CLEO-PR and OECC/PS), Kyoto, Japan, 2013, paper WR2-5.

E. Iannoe, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

H. Terauchi, Y. Matsuda, A. Toyota, and A. Maruta, “Noise tolerance of eigenvalue modulated optical transmission system based on digital coherent technology,” in 19th OptoElectronics and Communications Conference/Australian Conference on Optical Fibre Technology (OECC/ACOF), Melbourne, Australia, 2014, p. 542.

Y. Matsuda, H. Terauchi, and A. Maruta, “Design of eigenvalue-multiplexed multi-level modulation optical transmission system,” in 19th OptoElectronics and Communications Conference/Australian Conference on Optical Fibre Technology (OECC/ACOF), Melbourne, Australia, 2014, paper TH12B3.

A. Maruta and Y. Matsuda, “Polarization division multiplexed optical eigenvalue modulation,” in International Conference on Photonics in Switching (PS), Florence, Italy, 2015, pp. 265–267.

A. Maruta, A. Toyota, Y. Matsuda, and Y. Ikeda, “Experimental demonstration of long haul transmission of eigenvalue modulated signals,” Tyrrhenian International Workshop on Digital Comminications (TIWDC), Florence, Italy, 2015, pp. 28–30.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

W. K. McClary, “Fast seismic inversion,” Geophysics 48, 1371–1372 (1983).

E. Iannoe, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).

[Crossref]

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).

[Crossref]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).

[Crossref]

I. T. Lima, V. S. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Computational complexity of nonlinear transforms applied to optical communications systems with normal dispersion fibers,” IEEE Photonics Conference (IPC), Reston, VA, 2015, pp. 277–278.

E. Meron, M. Feder, and M. Shtaif, “On the achievable communication rates of generalized soliton transmission systems,” arXiv:1207.0297 (2012).

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature 411, 1027–1030 (2001).

P. Kazakopoulos and A. L. Moustakas, “On the soliton spectral efficiency in nonlinear optical fibers,” in IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, 2016, pp. 610–614.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).

[Crossref]

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lawery, “Digital fiber nonlinearity compensation: toward 1-Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).

[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).

[Crossref]

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London A 361, 413–446 (1978).

N. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

[Crossref]

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11, 395–399 (1993).

[Crossref]

I. T. Lima, V. S. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Computational complexity of nonlinear transforms applied to optical communications systems with normal dispersion fibers,” IEEE Photonics Conference (IPC), Reston, VA, 2015, pp. 277–278.

S. Oda, A. Maruta, and K. Kitayama, “All-optical quantization scheme based on fiber nonlinearity,” IEEE Photon. Technol. Lett. 16, 587–589 (2004).

[Crossref]

T. Trogdon and S. Olver, “Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations,” Proc. R. Soc. London A 469, 20120330 (2012).

[Crossref]

G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Comput. Phys. 102, 252–264 (1992).

[Crossref]

A. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform, 1st ed. (Academic, 2010).

A. R. Osborne, “The hyperelliptic inverse scattering transform for the periodic, defocusing nonlinear Schrödinger equation,” J. Comput. Phys. 109, 93–107 (1993).

[Crossref]

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper Tu2A.1.

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.B.2.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

M. Tan, P. Rosa, I. D. Phillips, and P. Harper, “Long-haul transmission performance evaluation of ultra-long Raman fiber laser based amplification influenced by second order co-pumping,” in Asia Communications and Photonics Conference, OSA Technical Digest (Optical Society of America, 2014), paper ATh1E.4.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

M. S. Pinsker, Information and Informational Stability of Random Variables and Processes (Holden Day, 1964), pp. 160–201.

L. L. Frumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method for solving the direct Zakharov-Shabat scattering problem,” J. Opt. Soc. Am. B 32, 290–295 (2015).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method of the fiber Bragg grating synthesis,” J. Opt. Soc. Am. B 24, 1451–1457 (2007).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, “Finite Bragg grating synthesis by numerical solution of Hermitian Gel’fand-Levitan-Marchenko equations,” J. Opt. Soc. Am. B 23, 2040–2045 (2006).

[Crossref]

L. Poladian, “Iterative and noniterative design algorithms for Bragg gratings,” Opt. Fiber Technol. 5, 215–222 (1999).

[Crossref]

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory 61, 6957–6974 (2015).

[Crossref]

S. Wahls and H. V. Poor, “Inverse nonlinear Fourier transforms via interpolation: the Ablowitz-Ladik case,” in International Symposium on Mathematical Theory of Networks and Systems (MTNS), Groningen, The Netherlands, 2014, pp. 1848–1855.

S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multisolitons in optical fiber,” in IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, 2015, pp. 1676–1680.

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vancouver, Canada, 2013, pp. 5780–5784.

S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, and S. K. Turitsyn, “Digital backpropagation in the nonlinear Fourier domain,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 445–449.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper Tu2A.1.

S. T. Le, J. E. Prilepsky, P. Rosa, J. D. Ania-Castanon, and S. K. Turitsyn, “Nonlinear inverse synthesis for optical links with distributed Raman amplification,” J. Lightwave Technol. 34, 1778–1786 (2016).

[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part II: eigenvalue communication,” Opt. Express 24, 18370–18381 (2016).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part I: theory and numerical methods,” Opt. Express 24, 18353–18369 (2016).

[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).

[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22, 26720–26741 (2014).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett. 113, 013901 (2014).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: linearization of the lossless fiber channel,” Opt. Express 21, 24344–24367 (2013).

[Crossref]

S. A. Derevyanko and J. E. Prilepsky, “Random input problem for the nonlinear Schrödinger equation,” Phys. Rev. E 78, 046610 (2008).

[Crossref]

J. E. Prilepsky and S. A. Derevyanko, “Breakup of a multisoliton state of the linearly damped nonlinear Schrödinger equation,” Phys. Rev. E 75, 036616 (2007).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasi-lossless and lossy optical fiber spans,” J. Opt. Soc. Am. B 24, 1254–1261 (2007).

[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and D. A. Yakushev, “Statistics of a noise-driven Manakov soliton,” J. Phys. A 39, 1297–1309 (2006).

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.B.2.

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

S. T. Le, S. Wahls, D. Lavery, J. E. Prilepsky, and S. K. Turitsyn, “Reduced complexity nonlinear inverse synthesis for nonlinearity compensation in optical fiber links,” in Proceedings of Conference on Lasers and Electro-Optics/Europe and the European Quantum Electronics Conference (CLEO/Europe-EQEC), Munich, Germany, 2015, paper CI_3_2.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform based transmissions with high order QAM formats,” in European Conference on Optical Communications (ECOC), Germany, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Optical communication based on the periodic nonlinear Fourier transform signal processing,” in IEEE 6th International Conference on Photonics (ICP), Kuching, Malaysia, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform based optical communication systems in a band-limited regime,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), Vancouver, Canada, 2016, paper JTu4A.34.

N. A. Shevchenko, J. E. Prilepsky, S. A. Derevyanko, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the per soliton capacity of the nonlinear optical fibre channel,” in IEEE Information Theory Workshop (ITW), Jeju Island, Korea, 2015, pp. 104–108.

S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, and S. K. Turitsyn, “Digital backpropagation in the nonlinear Fourier domain,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 445–449.

J. E. Prilepsky and S. K. Turitsyn, “Eigenvalue communications in nonlinear fiber channels,” in Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications, K. Porsezian and R. Ganapathy, eds. (CRC Press, 2015) Chap. 18, pp. 459–490.

J. G. Proakis, Digital Communications (McGraw-Hill, 2001).

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).

[Crossref]

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).

[Crossref]

D. Rafique, “Fiber nonlinearity compensation: commercial applications and complexity analysis,” J. Lightwave Technol. 34, 544–553 (2016).

[Crossref]

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lawery, “Digital fiber nonlinearity compensation: toward 1-Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).

[Crossref]

F. Ahmad and M. Razzagh, “A numerical solution to the Gel’fand-Levitan-Marchenko equation,” Appl. Math. Comput. 89, 31–39 (1998).

D. J. Richardson, “New optical fibres for high-capacity optical communications,” Philos. Trans. R. Soc. A 374, 20140441 (2016).

[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).

[Crossref]

D. J. Richardson, “Filling the light pipe,” Science 330, 327–328 (2010).

[Crossref]

A. Arico, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo 48, 75–88 (2011).

[Crossref]

S. T. Le, J. E. Prilepsky, P. Rosa, J. D. Ania-Castanon, and S. K. Turitsyn, “Nonlinear inverse synthesis for optical links with distributed Raman amplification,” J. Lightwave Technol. 34, 1778–1786 (2016).

[Crossref]

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

M. Tan, P. Rosa, I. D. Phillips, and P. Harper, “Long-haul transmission performance evaluation of ultra-long Raman fiber laser based amplification influenced by second order co-pumping,” in Asia Communications and Photonics Conference, OSA Technical Digest (Optical Society of America, 2014), paper ATh1E.4.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).

[Crossref]

R. J. Essiambre, R. W. Tkach, and R. Ryf, “Fiber nonlinearity and capacity: single mode and multimode fibers,” in Optical Fiber Telecommunications, I. Kaminow, T. Li, and A. E. Willner, eds., 6th ed., Vol. VIB of Systems and Networks (Academic, 2013), Chap. 1, pp. 1–43.

I. Tavakkolnia and M. Safari, “Signalling over nonlinear fibre-optic channels by utilizing both solitonic and radiative spectra,” in IEEE European Conference on Networks and Communications (EuCNC), Paris, France, 2015, pp. 103–107.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).

[Crossref]

B. Deconinck, H. Heil, A. Bobenko, M. Van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Math. Comput. 73, 1417–1443 (2004).

[Crossref]

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.2.

A. Arico, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo 48, 75–88 (2011).

[Crossref]

L. Fermo, C. van der Mee, and S. Seatzu, “Numerical solution of the direct scattering problem for the nonlinear Schrödinger equation,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 5–8.

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 13–16.

E. Forestieri and M. Secondini, “The nonlinear fiber-optic channel: modeling and achievable information rate,” in Progress in Electromagnetics Research Symposium (PIERS), Prague, Czech Republic, 2015, pp. 1276–1283.

N. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

[Crossref]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, 1981).

E. Iannoe, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. J. Exp. Theor. Phys. 34, 62–69 (1972).

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

[Crossref]

L. L. Frumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method for solving the direct Zakharov-Shabat scattering problem,” J. Opt. Soc. Am. B 32, 290–295 (2015).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method of the fiber Bragg grating synthesis,” J. Opt. Soc. Am. B 24, 1451–1457 (2007).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, “Finite Bragg grating synthesis by numerical solution of Hermitian Gel’fand-Levitan-Marchenko equations,” J. Opt. Soc. Am. B 23, 2040–2045 (2006).

[Crossref]

P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).

[Crossref]

N. A. Shevchenko, J. E. Prilepsky, S. A. Derevyanko, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the per soliton capacity of the nonlinear optical fibre channel,” in IEEE Information Theory Workshop (ITW), Jeju Island, Korea, 2015, pp. 104–108.

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

E. Meron, M. Feder, and M. Shtaif, “On the achievable communication rates of generalized soliton transmission systems,” arXiv:1207.0297 (2012).

J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).

[Crossref]

J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete-scattering algorithms,” J. Lightwave Technol. 21, 254–263 (2003).

[Crossref]

J. Skaar and R. Feced, “Reconstruction of gratings from noisy reflection data,” J. Opt. Soc. Am. A 19, 2229–2237 (2002).

[Crossref]

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).

[Crossref]

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lawery, “Digital fiber nonlinearity compensation: toward 1-Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).

[Crossref]

S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).

[Crossref]

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature 411, 1027–1030 (2001).

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).

[Crossref]

C. Swierczewski and B. Deconinck, “Computing Riemann theta functions in Sage with applications,” Math. Comput. Simul. 127, 263–272 (2016).

[Crossref]

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

M. Tan, P. Rosa, I. D. Phillips, and P. Harper, “Long-haul transmission performance evaluation of ultra-long Raman fiber laser based amplification influenced by second order co-pumping,” in Asia Communications and Photonics Conference, OSA Technical Digest (Optical Society of America, 2014), paper ATh1E.4.

I. Tavakkolnia and M. Safari, “Signalling over nonlinear fibre-optic channels by utilizing both solitonic and radiative spectra,” in IEEE European Conference on Networks and Communications (EuCNC), Paris, France, 2015, pp. 103–107.

H. Terauchi and A. Maruta, “Eigenvalue modulated optical transmission system based on digital coherent technology,” in 10th Conference on Lasers and Electro-Optics Pacific Rim, and the 18th OptoElectronics and Communications Conference/Photonics in Switching (CLEO-PR and OECC/PS), Kyoto, Japan, 2013, paper WR2-5.

H. Terauchi, Y. Matsuda, A. Toyota, and A. Maruta, “Noise tolerance of eigenvalue modulated optical transmission system based on digital coherent technology,” in 19th OptoElectronics and Communications Conference/Australian Conference on Optical Fibre Technology (OECC/ACOF), Melbourne, Australia, 2014, p. 542.

Y. Matsuda, H. Terauchi, and A. Maruta, “Design of eigenvalue-multiplexed multi-level modulation optical transmission system,” in 19th OptoElectronics and Communications Conference/Australian Conference on Optical Fibre Technology (OECC/ACOF), Melbourne, Australia, 2014, paper TH12B3.

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147, 166–186 (1998).

[Crossref]

A. R. C. X. Liu, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7, 560–568 (2013).

[Crossref]

C. Xi, L. Xiang, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Technical Digest of Optical Fiber Communication Conference and Exposition (OFC/NFOEC) and the National Fiber Optic Engineers Conference (2012), paper OTh3C.1.

R. J. Essiambre, R. W. Tkach, and R. Ryf, “Fiber nonlinearity and capacity: single mode and multimode fibers,” in Optical Fiber Telecommunications, I. Kaminow, T. Li, and A. E. Willner, eds., 6th ed., Vol. VIB of Systems and Networks (Academic, 2013), Chap. 1, pp. 1–43.

A. Toyota and A. Maruta, “Wavelength division multiplexed optical eigenvalue modulated system,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 43–45.

A. Maruta, A. Toyota, Y. Matsuda, and Y. Ikeda, “Experimental demonstration of long haul transmission of eigenvalue modulated signals,” Tyrrhenian International Workshop on Digital Comminications (TIWDC), Florence, Italy, 2015, pp. 28–30.

H. Terauchi, Y. Matsuda, A. Toyota, and A. Maruta, “Noise tolerance of eigenvalue modulated optical transmission system based on digital coherent technology,” in 19th OptoElectronics and Communications Conference/Australian Conference on Optical Fibre Technology (OECC/ACOF), Melbourne, Australia, 2014, p. 542.

E. R. Tracy and H. H. Chen, “Nonlinear self-modulation: an exactly solvable model,” Phys. Rev. A 37, 815–839 (1988).

[Crossref]

W. F. Trench, “An algorithm for the inversion of finite Toeplitz matrices,” J. Soc. Ind. Appl. Math. 12, 515–522 (1964).

[Crossref]

T. Trogdon and S. Olver, “Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations,” Proc. R. Soc. London A 469, 20120330 (2012).

[Crossref]

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper Tu2A.1.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform based optical communication systems in a band-limited regime,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), Vancouver, Canada, 2016, paper JTu4A.34.

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part I: theory and numerical methods,” Opt. Express 24, 18353–18369 (2016).

[Crossref]

S. T. Le, J. E. Prilepsky, P. Rosa, J. D. Ania-Castanon, and S. K. Turitsyn, “Nonlinear inverse synthesis for optical links with distributed Raman amplification,” J. Lightwave Technol. 34, 1778–1786 (2016).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part II: eigenvalue communication,” Opt. Express 24, 18370–18381 (2016).

[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).

[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22, 26720–26741 (2014).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett. 113, 013901 (2014).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: linearization of the lossless fiber channel,” Opt. Express 21, 24344–24367 (2013).

[Crossref]

E. G. Turitsyna and S. K. Turitsyn, “Digital signal processing based on inverse scattering transform,” Opt. Lett. 38, 4186–4188 (2013).

[Crossref]

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101, 123903 (2008).

[Crossref]

S. K. Turitsyn and S. A. Derevyanko, “Soliton-based discriminator of noncoherent optical pulses,” Phys. Rev. A 78, 063819 (2008).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasi-lossless and lossy optical fiber spans,” J. Opt. Soc. Am. B 24, 1254–1261 (2007).

[Crossref]

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).

[Crossref]

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Fokker-Planck equation approach to the description of soliton statistics in optical fiber transmission systems,” J. Opt. Soc. Am. B 22, 743–752 (2005).

[Crossref]

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Non-Gaussian statistics of an optical soliton in the presence of amplified spontaneous emission,” Opt. Lett. 28, 2097–2099 (2003).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.B.2.

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

J. E. Prilepsky and S. K. Turitsyn, “Eigenvalue communications in nonlinear fiber channels,” in Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications, K. Porsezian and R. Ganapathy, eds. (CRC Press, 2015) Chap. 18, pp. 459–490.

S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, and S. K. Turitsyn, “Digital backpropagation in the nonlinear Fourier domain,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 445–449.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Optical communication based on the periodic nonlinear Fourier transform signal processing,” in IEEE 6th International Conference on Photonics (ICP), Kuching, Malaysia, 2016, pp. 1–3.

N. A. Shevchenko, J. E. Prilepsky, S. A. Derevyanko, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the per soliton capacity of the nonlinear optical fibre channel,” in IEEE Information Theory Workshop (ITW), Jeju Island, Korea, 2015, pp. 104–108.

S. T. Le, S. Wahls, D. Lavery, J. E. Prilepsky, and S. K. Turitsyn, “Reduced complexity nonlinear inverse synthesis for nonlinearity compensation in optical fiber links,” in Proceedings of Conference on Lasers and Electro-Optics/Europe and the European Quantum Electronics Conference (CLEO/Europe-EQEC), Munich, Germany, 2015, paper CI_3_2.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform based transmissions with high order QAM formats,” in European Conference on Optical Communications (ECOC), Germany, 2016, pp. 1–3.

S. Wahls and V. Vaibhav, “Introducing the fast inverse NFT,” in Optical Fiber Communication Conference (OFC) (2017, to appear).

S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” arXiv:1607.01305v2 [cs.IT] (2016).

V. Vaibhav and S. Wahls, “Multipoint Newton-type nonlinear Fourier transform for detecting multi-solitons,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper W2A.34.

S. Wahls and V. Vaibhav, “Fast generation of multi-solitons using the Darboux transform,” in Munich Workshop on Information Theory of Optical Fiber (MIO), Germany, 2015.

S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).

[Crossref]

L. Fermo, C. van der Mee, and S. Seatzu, “Numerical solution of the direct scattering problem for the nonlinear Schrödinger equation,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 5–8.

B. Deconinck, H. Heil, A. Bobenko, M. Van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Math. Comput. 73, 1417–1443 (2004).

[Crossref]

J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, “Fast and backward stable computation of roots of polynomials,” SIAM J. Matrix Anal. Appl. 36, 942–973 (2015).

[Crossref]

J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).

[Crossref]

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory 61, 6957–6974 (2015).

[Crossref]

S. T. Le, S. Wahls, D. Lavery, J. E. Prilepsky, and S. K. Turitsyn, “Reduced complexity nonlinear inverse synthesis for nonlinearity compensation in optical fiber links,” in Proceedings of Conference on Lasers and Electro-Optics/Europe and the European Quantum Electronics Conference (CLEO/Europe-EQEC), Munich, Germany, 2015, paper CI_3_2.

S. Wahls and H. V. Poor, “Inverse nonlinear Fourier transforms via interpolation: the Ablowitz-Ladik case,” in International Symposium on Mathematical Theory of Networks and Systems (MTNS), Groningen, The Netherlands, 2014, pp. 1848–1855.

S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multisolitons in optical fiber,” in IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, 2015, pp. 1676–1680.

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vancouver, Canada, 2013, pp. 5780–5784.

S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, and S. K. Turitsyn, “Digital backpropagation in the nonlinear Fourier domain,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 445–449.

S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” arXiv:1607.01305v2 [cs.IT] (2016).

V. Vaibhav and S. Wahls, “Multipoint Newton-type nonlinear Fourier transform for detecting multi-solitons,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper W2A.34.

S. Wahls and V. Vaibhav, “Fast generation of multi-solitons using the Darboux transform,” in Munich Workshop on Information Theory of Optical Fiber (MIO), Germany, 2015.

S. Wahls and V. Vaibhav, “Introducing the fast inverse NFT,” in Optical Fiber Communication Conference (OFC) (2017, to appear).

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).

[Crossref]

J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, “Fast and backward stable computation of roots of polynomials,” SIAM J. Matrix Anal. Appl. 36, 942–973 (2015).

[Crossref]

P. J. Winzer, “Scaling optical fiber networks: challenges and solutions,” Opt. Photon. News 26(3), 28–35 (2015).

P. J. Winzer, “Spatial multiplexing in fiber optics: the 10X scaling of metro/core capacities,” Bell Labs Tech. J. 19, 22–30 (2014).

A. R. C. X. Liu, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7, 560–568 (2013).

[Crossref]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010).

[Crossref]

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).

[Crossref]

C. Xi, L. Xiang, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Technical Digest of Optical Fiber Communication Conference and Exposition (OFC/NFOEC) and the National Fiber Optic Engineers Conference (2012), paper OTh3C.1.

C. Xi, L. Xiang, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Technical Digest of Optical Fiber Communication Conference and Exposition (OFC/NFOEC) and the National Fiber Optic Engineers Conference (2012), paper OTh3C.1.

G. B. Xiao and K. Yashiro, “An efficient algorithm for solving Zakharov-Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 50, 807–811 (2002).

[Crossref]

P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).

[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and D. A. Yakushev, “Statistics of a noise-driven Manakov soliton,” J. Phys. A 39, 1297–1309 (2006).

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Fokker-Planck equation approach to the description of soliton statistics in optical fiber transmission systems,” J. Opt. Soc. Am. B 22, 743–752 (2005).

[Crossref]

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Non-Gaussian statistics of an optical soliton in the presence of amplified spontaneous emission,” Opt. Lett. 28, 2097–2099 (2003).

[Crossref]

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).

M. I. Yousefi and X. Yangzhang, “Linear and nonlinear frequency-division multiplexing,” arXiv:1207.0297 (2016).

G. B. Xiao and K. Yashiro, “An efficient algorithm for solving Zakharov-Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 50, 807–811 (2002).

[Crossref]

S. Hari, F. Kschischang, and M. Yousefi, “Multi-eigenvalue communication via the nonlinear Fourier transform,” in 27th Biennial Symposium on Communications (QBSC), Ontario, Canada, 2014, pp. 92–95.

S. Hari, M. I. Yousefi, and F. R. Kschischang, “Multieigenvalue communication,” J. Lightwave Technol. 34, 3110–3117 (2016).

[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: mathematical tools,” IEEE Trans. Inf. Theory 60, 4312–4328 (2014).

[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: numerical methods,” IEEE Trans. Inf. Theory 60, 4329–4345 (2014).

[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: spectrum modulation,” IEEE Trans. Inf. Theory 60, 4346–4369 (2014).

[Crossref]

M. I. Yousefi and X. Yangzhang, “Linear and nonlinear frequency-division multiplexing,” arXiv:1207.0297 (2016).

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. J. Exp. Theor. Phys. 34, 62–69 (1972).

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).

[Crossref]

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).

[Crossref]

Q. Zhang and T. H. Chan, “A Gaussian noise model of spectral amplitudes in soliton communication systems,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 455–459.

Q. Zhang and T. H. Chan, “A spectral domain noise model for optical fibre channels,” in IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, 2015, paper We-AM2-9.1.

Q. Zhang and T. H. Chan, “Achievable rates of soliton communication systems,” in IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, 2016, pp. 605–609.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

C. Xi, L. Xiang, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Technical Digest of Optical Fiber Communication Conference and Exposition (OFC/NFOEC) and the National Fiber Optic Engineers Conference (2012), paper OTh3C.1.

S. Zohar, “Toeplitz matrix inversion: the algorithm of W. F. Trench,” J. Assoc. Comput. Mach. 16, 592–601 (1969).

[Crossref]

F. Ahmad and M. Razzagh, “A numerical solution to the Gel’fand-Levitan-Marchenko equation,” Appl. Math. Comput. 89, 31–39 (1998).

P. J. Winzer, “Spatial multiplexing in fiber optics: the 10X scaling of metro/core capacities,” Bell Labs Tech. J. 19, 22–30 (2014).

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

[Crossref]

A. Arico, G. Rodriguez, and S. Seatzu, “Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data,” Calcolo 48, 75–88 (2011).

[Crossref]

V. Kotlyarov and A. Its, “Explicit formulas for the solutions of a nonlinear Schrodinger equation,” Doklady Akad. Nauk Ukrainian SSR Ser. A 10, 965–968 (1976), http://arxiv.org/abs/1401.4445v1 .

W. K. McClary, “Fast seismic inversion,” Geophysics 48, 1371–1372 (1983).

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).

[Crossref]

J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).

[Crossref]

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).

[Crossref]

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).

[Crossref]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).

[Crossref]

S. Oda, A. Maruta, and K. Kitayama, “All-optical quantization scheme based on fiber nonlinearity,” IEEE Photon. Technol. Lett. 16, 587–589 (2004).

[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P. T. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).

[Crossref]

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lawery, “Digital fiber nonlinearity compensation: toward 1-Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).

[Crossref]

P. Frangos and D. Jaggard, “A numerical solution to the Zakharov-Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).

[Crossref]

G. B. Xiao and K. Yashiro, “An efficient algorithm for solving Zakharov-Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 50, 807–811 (2002).

[Crossref]

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).

[Crossref]

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory 61, 6957–6974 (2015).

[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part II: numerical methods,” IEEE Trans. Inf. Theory 60, 4329–4345 (2014).

[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part III: spectrum modulation,” IEEE Trans. Inf. Theory 60, 4346–4369 (2014).

[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, Part I: mathematical tools,” IEEE Trans. Inf. Theory 60, 4312–4328 (2014).

[Crossref]

P. Boito, Y. Eidelman, L. Gemignami, and I. Gohberg, “Implicit QR with compression,” Indag. Math. 23, 733–761 (2012).

[Crossref]

S. Zohar, “Toeplitz matrix inversion: the algorithm of W. F. Trench,” J. Assoc. Comput. Mach. 16, 592–601 (1969).

[Crossref]

A. R. Osborne, “The hyperelliptic inverse scattering transform for the periodic, defocusing nonlinear Schrödinger equation,” J. Comput. Phys. 109, 93–107 (1993).

[Crossref]

G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Comput. Phys. 102, 252–264 (1992).

[Crossref]

S. Burtsev, R. Camassa, and I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147, 166–186 (1998).

[Crossref]

S. Hari and F. R. Kschischang, “Bi-directional algorithm for computing discrete spectral amplitudes in the NFT,” J. Lightwave Technol. 34, 3529–3537 (2016).

[Crossref]

S. T. Le, J. E. Prilepsky, P. Rosa, J. D. Ania-Castanon, and S. K. Turitsyn, “Nonlinear inverse synthesis for optical links with distributed Raman amplification,” J. Lightwave Technol. 34, 1778–1786 (2016).

[Crossref]

S. T. Le, I. D. Philips, J. E. Prilepsky, P. Harper, A. D. Ellis, and S. K. Turitsyn, “Demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” J. Lightwave Technol. 34, 2459–2466 (2016).

[Crossref]

J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete-scattering algorithms,” J. Lightwave Technol. 21, 254–263 (2003).

[Crossref]

S. Hari, M. I. Yousefi, and F. R. Kschischang, “Multieigenvalue communication,” J. Lightwave Technol. 34, 3110–3117 (2016).

[Crossref]

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11, 395–399 (1993).

[Crossref]

H. Bülow, “Experimental demonstration of optical signal detection using nonlinear Fourier transform,” J. Lightwave Technol. 33, 1433–1439 (2015).

[Crossref]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010).

[Crossref]

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the nonlinear Shannon limit,” J. Lightwave Technol. 28, 423–433 (2010).

[Crossref]

E. Ip and J. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008).

[Crossref]

S. L. Jansen, D. Van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).

[Crossref]

D. Rafique, “Fiber nonlinearity compensation: commercial applications and complexity analysis,” J. Lightwave Technol. 34, 544–553 (2016).

[Crossref]

M. J. Ablowitz and J. F. Ladik, “Nonlinear differential-difference equations,” J. Math. Phys. 16, 598–603 (1975).

[Crossref]

R. I. Killey and C. Behrens, “Shannon’s theory in nonlinear systems,” J. Mod. Opt. 58, 1–10 (2011).

[Crossref]

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Fokker-Planck equation approach to the description of soliton statistics in optical fiber transmission systems,” J. Opt. Soc. Am. B 22, 743–752 (2005).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, “Finite Bragg grating synthesis by numerical solution of Hermitian Gel’fand-Levitan-Marchenko equations,” J. Opt. Soc. Am. B 23, 2040–2045 (2006).

[Crossref]

O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method of the fiber Bragg grating synthesis,” J. Opt. Soc. Am. B 24, 1451–1457 (2007).

[Crossref]

L. L. Frumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, “Efficient numerical method for solving the direct Zakharov-Shabat scattering problem,” J. Opt. Soc. Am. B 32, 290–295 (2015).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Conversion of a chirped Gaussian pulse to a soliton or a bound multisoliton state in quasi-lossless and lossy optical fiber spans,” J. Opt. Soc. Am. B 24, 1254–1261 (2007).

[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and D. A. Yakushev, “Statistics of a noise-driven Manakov soliton,” J. Phys. A 39, 1297–1309 (2006).

W. F. Trench, “An algorithm for the inversion of finite Toeplitz matrices,” J. Soc. Ind. Appl. Math. 12, 515–522 (1964).

[Crossref]

J. Frauendiener and C. Klein, “Computational approach to hyperelliptic Riemann surfaces,” Lett. Math. Phys. 105, 379–400 (2015).

[Crossref]

B. Deconinck, H. Heil, A. Bobenko, M. Van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Math. Comput. 73, 1417–1443 (2004).

[Crossref]

C. Swierczewski and B. Deconinck, “Computing Riemann theta functions in Sage with applications,” Math. Comput. Simul. 127, 263–272 (2016).

[Crossref]

S. A. Derevyanko, J. E. Prilepsky, and S. K. Turitsyn, “Capacity estimates for optical transmission based on the nonlinear Fourier transform,” Nat. Commun. 7, 12710 (2016).

[Crossref]

A. R. C. X. Liu, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7, 560–568 (2013).

[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).

[Crossref]

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature 411, 1027–1030 (2001).

J. D. Ania-Castañón, “Quasi-lossless transmission using second-order Raman amplification and fibre Bragg gratings,” Opt. Express 12, 4372–4377 (2004).

[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22, 26720–26741 (2014).

[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: linearization of the lossless fiber channel,” Opt. Express 21, 24344–24367 (2013).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part I: theory and numerical methods,” Opt. Express 24, 18353–18369 (2016).

[Crossref]

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform for fiber-optic communications, Part II: eigenvalue communication,” Opt. Express 24, 18370–18381 (2016).

[Crossref]

A. Buryak, J. Bland-Hawthorn, and V. Steblina, “Comparison of inverse scattering algorithms for designing ultrabroadband fiber Bragg gratings,” Opt. Express 17, 1995–2004 (2009).

[Crossref]

L. Poladian, “Iterative and noniterative design algorithms for Bragg gratings,” Opt. Fiber Technol. 5, 215–222 (1999).

[Crossref]

S. A. Derevyanko, “Design of a flat-top fiber Bragg filter via quasi-random modulation of the refractive index,” Opt. Lett. 33, 2404–2406 (2008).

[Crossref]

E. G. Turitsyna and S. K. Turitsyn, “Digital signal processing based on inverse scattering transform,” Opt. Lett. 38, 4186–4188 (2013).

[Crossref]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).

[Crossref]

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).

[Crossref]

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).

[Crossref]

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” Opt. Lett. 15, 314–316 (1990).

[Crossref]

S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Non-Gaussian statistics of an optical soliton in the presence of amplified spontaneous emission,” Opt. Lett. 28, 2097–2099 (2003).

[Crossref]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).

[Crossref]

P. J. Winzer, “Scaling optical fiber networks: challenges and solutions,” Opt. Photon. News 26(3), 28–35 (2015).

D. J. Richardson, “New optical fibres for high-capacity optical communications,” Philos. Trans. R. Soc. A 374, 20140441 (2016).

[Crossref]

P. Bayvel, R. Maher, T. Xu, G. Liga, N. A. Shevchenko, D. Lavery, A. Alvarado, and R. I. Killey, “Maximizing the optical network capacity,” Philos. Trans. R. Soc. A 374, 20140440 (2016).

[Crossref]

E. Agrell, A. Alvarado, and F. R. Kschishang, “Implications of information theory in optical fibre communications,” Philos. Trans. R. Soc. A 374, 20140438 (2016).

[Crossref]

S. K. Turitsyn and S. A. Derevyanko, “Soliton-based discriminator of noncoherent optical pulses,” Phys. Rev. A 78, 063819 (2008).

[Crossref]

E. R. Tracy and H. H. Chen, “Nonlinear self-modulation: an exactly solvable model,” Phys. Rev. A 37, 815–839 (1988).

[Crossref]

J. E. Prilepsky and S. A. Derevyanko, “Breakup of a multisoliton state of the linearly damped nonlinear Schrödinger equation,” Phys. Rev. E 75, 036616 (2007).

[Crossref]

S. A. Derevyanko and J. E. Prilepsky, “Random input problem for the nonlinear Schrödinger equation,” Phys. Rev. E 78, 046610 (2008).

[Crossref]

S. A. Derevyanko, “Appearance of bound states in random potentials with applications to soliton theory,” Phys. Rev. E 84, 016601 (2011).

[Crossref]

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett. 101, 163901 (2008).

[Crossref]

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels,” Phys. Rev. Lett. 113, 013901 (2014).

[Crossref]

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, “Ultralong Raman fibre lasers as virtually lossless optical media,” Phys. Rev. Lett. 96, 023902 (2006).

[Crossref]

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101, 123903 (2008).

[Crossref]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).

T. Trogdon and S. Olver, “Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations,” Proc. R. Soc. London A 469, 20120330 (2012).

[Crossref]

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London A 361, 413–446 (1978).

D. J. Richardson, “Filling the light pipe,” Science 330, 327–328 (2010).

[Crossref]

D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM J. Appl. Math. 31, 121–133 (1976).

[Crossref]

J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, “Fast and backward stable computation of roots of polynomials,” SIAM J. Matrix Anal. Appl. 36, 942–973 (2015).

[Crossref]

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. J. Exp. Theor. Phys. 34, 62–69 (1972).

S. V. Manakov, “On the theory of two-dimensional stationary self focussing of electromagnetic waves,” Sov. Phys. J. Exp. Theor. Phys. 38, 248–253 (1974).

N. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

[Crossref]

M. J. Ablowitz and J. F. Ladik, “A difference scheme and inverse scattering,” Stud. Appl. Math. 55, 213–229 (1976).

[Crossref]

Y. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Stud. Appl. Math. 65, 113–158 (1981).

[Crossref]

L. Fermo, C. van der Mee, and S. Seatzu, “Numerical solution of the direct scattering problem for the nonlinear Schrödinger equation,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 5–8.

A. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform, 1st ed. (Academic, 2010).

A. Bobenko and C. Klein, eds., Computational Approach to Riemann Surfaces (Springer, 2011), Vol. 2013.

B. Deconinck, M. S. Patterson, and C. Swierczewski, Computing the Riemann constant vector, 2015, preprint, https://depts.washington.edu/bdecon/papers/pdfs/rcv.pdf .

S. Wahls and H. V. Poor, “Introducing the fast nonlinear Fourier transform,” in International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vancouver, Canada, 2013, pp. 5780–5784.

M. Kamalian, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Periodic nonlinear Fourier transform based transmissions with high order QAM formats,” in European Conference on Optical Communications (ECOC), Germany, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. K. Turitsyn, “Optical communication based on the periodic nonlinear Fourier transform signal processing,” in IEEE 6th International Conference on Photonics (ICP), Kuching, Malaysia, 2016, pp. 1–3.

M. Kamalian Kopae, J. E. Prilepsky, S. T. Le, and S. Turitsyn, “Periodic nonlinear Fourier transform based optical communication systems in a band-limited regime,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), Vancouver, Canada, 2016, paper JTu4A.34.

S. T. Le, I. Philips, J. Prilepsky, P. Harper, N. Doran, A. D. Ellis, and S. Turitsyn, “First experimental demonstration of nonlinear inverse synthesis transmission over transoceanic distances,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper Tu2A.1.

I. Tavakkolnia and M. Safari, “Signalling over nonlinear fibre-optic channels by utilizing both solitonic and radiative spectra,” in IEEE European Conference on Networks and Communications (EuCNC), Paris, France, 2015, pp. 103–107.

A. Aref, S. T. Le, and H. Buelow, “Demonstration of fully nonlinear spectrum modulated system in the highly nonlinear optical transmission regime,” in European Conference on Optical Communication (ECOC), Germany, 2016, paper Th.3.B.2.

S. Hari, F. Kschischang, and M. Yousefi, “Multi-eigenvalue communication via the nonlinear Fourier transform,” in 27th Biennial Symposium on Communications (QBSC), Ontario, Canada, 2014, pp. 92–95.

S. T. Le, S. Wahls, D. Lavery, J. E. Prilepsky, and S. K. Turitsyn, “Reduced complexity nonlinear inverse synthesis for nonlinearity compensation in optical fiber links,” in Proceedings of Conference on Lasers and Electro-Optics/Europe and the European Quantum Electronics Conference (CLEO/Europe-EQEC), Munich, Germany, 2015, paper CI_3_2.

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Achievable information rate of nonlinear inverse synthesis based 16QAM OFDM transmission,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Th.2.PS2.SC5.

H. Bülow, “Experimental assessment of nonlinear Fourier transformation based detection under fiber nonlinearity,” in European Conference on Optical Communications (ECOC), Cannes, France, 2014, paper We.2.3.2.

S. Wahls and H. V. Poor, “Inverse nonlinear Fourier transforms via interpolation: the Ablowitz-Ladik case,” in International Symposium on Mathematical Theory of Networks and Systems (MTNS), Groningen, The Netherlands, 2014, pp. 1848–1855.

S. Wahls and H. V. Poor, “Fast inverse nonlinear Fourier transform for generating multisolitons in optical fiber,” in IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, 2015, pp. 1676–1680.

V. Vaibhav and S. Wahls, “Multipoint Newton-type nonlinear Fourier transform for detecting multi-solitons,” in Optical Fiber Communication Conference (OFC), Anaheim, CA, 2016, paper W2A.34.

S. Wahls and V. Vaibhav, “Fast generation of multi-solitons using the Darboux transform,” in Munich Workshop on Information Theory of Optical Fiber (MIO), Germany, 2015.

S. Wahls and V. Vaibhav, “Fast inverse nonlinear Fourier transforms for continuous spectra of Zakharov-Shabat type,” arXiv:1607.01305v2 [cs.IT] (2016).

S. Wahls and V. Vaibhav, “Introducing the fast inverse NFT,” in Optical Fiber Communication Conference (OFC) (2017, to appear).

S. Civelli, L. Barletti, and M. Secondini, “Numerical methods for the inverse nonlinear Fourier transform,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 13–16.

E. Forestieri and M. Secondini, “The nonlinear fiber-optic channel: modeling and achievable information rate,” in Progress in Electromagnetics Research Symposium (PIERS), Prague, Czech Republic, 2015, pp. 1276–1283.

A. Span and S. ten Brink, “Optical communications using the nonlinear Fourier transform,” in Munich Workshop on Information Theory of Optical Fiber (MIO), Germany, 2015, http://bit.ly/2ePRyCZ .

H. Buelow, V. Aref, and W. Idler, “Transmission of waveform determined by 7 eigenvalues with PSK-modulated spectral amplitude,” in 42nd European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.E.2.

I. T. Lima, V. S. Grigoryan, M. O’sullivan, and C. R. Menyuk, “Computational complexity of nonlinear transforms applied to optical communications systems with normal dispersion fibers,” IEEE Photonics Conference (IPC), Reston, VA, 2015, pp. 277–278.

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.3.

H. Bülow, “Nonlinear Fourier transformation based coherent detection scheme for discrete spectrum,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2015, paper W3K.2.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st European Conference on Optical Communications (ECOC), Valencia, Spain, 2015, paper Tu 1.1.2.

H. Terauchi and A. Maruta, “Eigenvalue modulated optical transmission system based on digital coherent technology,” in 10th Conference on Lasers and Electro-Optics Pacific Rim, and the 18th OptoElectronics and Communications Conference/Photonics in Switching (CLEO-PR and OECC/PS), Kyoto, Japan, 2013, paper WR2-5.

H. Terauchi, Y. Matsuda, A. Toyota, and A. Maruta, “Noise tolerance of eigenvalue modulated optical transmission system based on digital coherent technology,” in 19th OptoElectronics and Communications Conference/Australian Conference on Optical Fibre Technology (OECC/ACOF), Melbourne, Australia, 2014, p. 542.

Y. Matsuda, H. Terauchi, and A. Maruta, “Design of eigenvalue-multiplexed multi-level modulation optical transmission system,” in 19th OptoElectronics and Communications Conference/Australian Conference on Optical Fibre Technology (OECC/ACOF), Melbourne, Australia, 2014, paper TH12B3.

A. Maruta, “Eigenvalue modulated optical transmission system (invited),” in 20th OptoElectronics and Communications Conference (OECC), Shanghai, China, 2015, paper JThA.21.

A. Toyota and A. Maruta, “Wavelength division multiplexed optical eigenvalue modulated system,” in Tyrrhenian International Workshop on Digital Communications (TIWDC), Florence, Italy, 2015, pp. 43–45.

A. Maruta, A. Toyota, Y. Matsuda, and Y. Ikeda, “Experimental demonstration of long haul transmission of eigenvalue modulated signals,” Tyrrhenian International Workshop on Digital Comminications (TIWDC), Florence, Italy, 2015, pp. 28–30.

A. Maruta and Y. Matsuda, “Polarization division multiplexed optical eigenvalue modulation,” in International Conference on Photonics in Switching (PS), Florence, Italy, 2015, pp. 265–267.

S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, and S. K. Turitsyn, “Digital backpropagation in the nonlinear Fourier domain,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 445–449.

Q. Zhang and T. H. Chan, “A Gaussian noise model of spectral amplitudes in soliton communication systems,” in IEEE 16th International Workshop in Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, 2015, pp. 455–459.

Q. Zhang and T. H. Chan, “A spectral domain noise model for optical fibre channels,” in IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, 2015, paper We-AM2-9.1.

E. Meron, M. Feder, and M. Shtaif, “On the achievable communication rates of generalized soliton transmission systems,” arXiv:1207.0297 (2012).

N. A. Shevchenko, J. E. Prilepsky, S. A. Derevyanko, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the per soliton capacity of the nonlinear optical fibre channel,” in IEEE Information Theory Workshop (ITW), Jeju Island, Korea, 2015, pp. 104–108.

M. Tan, P. Rosa, I. D. Phillips, and P. Harper, “Long-haul transmission performance evaluation of ultra-long Raman fiber laser based amplification influenced by second order co-pumping,” in Asia Communications and Photonics Conference, OSA Technical Digest (Optical Society of America, 2014), paper ATh1E.4.

R. J. Essiambre, R. W. Tkach, and R. Ryf, “Fiber nonlinearity and capacity: single mode and multimode fibers,” in Optical Fiber Telecommunications, I. Kaminow, T. Li, and A. E. Willner, eds., 6th ed., Vol. VIB of Systems and Networks (Academic, 2013), Chap. 1, pp. 1–43.

G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010).

M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication Systems and Networks (Artech House, 2013).

J. G. Proakis, Digital Communications (McGraw-Hill, 2001).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

E. Iannoe, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, 1981).

G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic, 2006).

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, 1995).

J. E. Prilepsky and S. K. Turitsyn, “Eigenvalue communications in nonlinear fiber channels,” in Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications, K. Porsezian and R. Ganapathy, eds. (CRC Press, 2015) Chap. 18, pp. 459–490.

I. D. Phillips, M. Tan, M. F. C. Stephens, M. E. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the Nonlinear-Shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference (OFC), Los Angeles, CA, 2014, paper M3C.1.

C. Xi, L. Xiang, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Technical Digest of Optical Fiber Communication Conference and Exposition (OFC/NFOEC) and the National Fiber Optic Engineers Conference (2012), paper OTh3C.1.

N. A. Shevchenko, S. A. Derevyanko, J. E. Prilepsky, A. Alvarado, P. Bavel, and S. K. Turitsyn, “A lower bound on the capacity of the noncentral chi channel with applications to soliton amplitude modulation,” arXiv:1609.02318 (2016).

Q. Zhang and T. H. Chan, “Achievable rates of soliton communication systems,” in IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, 2016, pp. 605–609.

P. Kazakopoulos and A. L. Moustakas, “On the soliton spectral efficiency in nonlinear optical fibers,” in IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, 2016, pp. 610–614.

M. S. Pinsker, Information and Informational Stability of Random Variables and Processes (Holden Day, 1964), pp. 160–201.

M. I. Yousefi and X. Yangzhang, “Linear and nonlinear frequency-division multiplexing,” arXiv:1207.0297 (2016).

S. T. Le, I. D. Philips, J. E. Prilepsky, M. Kamalian, A. D. Ellis, P. Harper, and S. K. Turitsyn, “Equalization-enhanced phase noise in nonlinear inverse synthesis transmissions,” in European Conference on Optical Communications (ECOC), Germany, 2016, paper Tu.3.B.2.