Abstract

We address the properties of fully three-dimensional solitons in complex parity-time (PT)-symmetric periodic lattices with focusing Kerr nonlinearity, and uncover that such lattices can stabilize both fundamental and vortex-carrying soliton states. The imaginary part of the lattice induces internal currents in the solitons that strongly affect their domains of existence and stability. The domain of stability for fundamental solitons can extend nearly up to the PT-symmetry breaking point, where the linear lattice spectrum becomes complex. Vortex solitons feature spatially asymmetric profiles in the PT-symmetric lattices, but they are found to still exist as stable states within narrow regions. Our results provide the first example of continuous families of stable three-dimensional propagating solitons supported by complex potentials.

© 2016 Optical Society of America

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Corrections

21 September 2016: A correction was made to the acknowledgment section.


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References

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2015 (3)

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

Y.-X. Chen and C.-Q. Dai, “Gaussian-type light bullets in power-law nonlinear media with PT-symmetric potential,” Opt. Commun. 338, 393–398 (2015).
[Crossref]

C. Hang and G.-X. Huang, “Weak-light solitons and their active control in a parity-time-symmetric atomic system,” Phys. Rev. A 91, 043833 (2015).

2014 (10)

C.-Q. Dai, X.-G. Wang, and G.-Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).

J. Yang, “Partially PT-symmetric optical potentials with all-real spectra and soliton families in multidimensions,” Opt. Lett. 39, 1133–1136 (2014).
[Crossref]

Z. Chen, J. Liu, S. Fu, Y. Li, and B. A. Malomed, “Discrete solitons and vortices on two-dimensional lattices of PT-symmetric couplers,” Opt. Express 22, 29679–29692 (2014).
[Crossref]

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT-symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

J. Lien, Y. Chen, N. Ishida, H. Chen, C. Hwang, and F. Nori, “Multistability and condensation of exciton-polaritons below threshold,” Phys. Rev. B 91, 024511 (2014).

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).

X. F. Zhu, H. Ramezani, C. Z. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4, 031042 (2014).
[Crossref]

L. Feng, Z. J. Wong, R. Ma, Y. Wang, and X. Zhang, “Single-mode laser by PT-symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

2013 (4)

C. Hang, G. Huang, and V. V. Konotop, “PT symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3, 041031 (2013).
[Crossref]

Y. V. Kartashov, “Vector solitons in parity-time-symmetric lattices,” Opt. Lett. 38, 2600–2603 (2013).
[Crossref]

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durécu, A. Couairon, and A. Mysyrowicz, “Self-guided propagation of ultrashort laser pulses in the anomalous dispersion region of transparent solids: a new regime of filamentation,” Phys. Rev. Lett. 110, 115003 (2013).
[Crossref]

2012 (2)

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).

C. Bersch, G. Onishchukov, and U. Peschel, “Optical gap solitons and truncated nonlinear Bloch waves in temporal lattices,” Phys. Rev. Lett. 109, 093903 (2012).
[Crossref]

2011 (3)

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805(R) (2011).

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, “Solitons in a chain of parity-time-invariant dimers,” Phys. Rev. E 84, 046609 (2011).

X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36, 2680–2682 (2011).
[Crossref]

2010 (7)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35, 2976–2978 (2010).
[Crossref]

S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-dimensional light bullets in arrays of waveguides,” Phys. Rev. Lett. 105, 263901 (2010).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg-Landau model with a two-dimensional transverse grating,” Phys. Rev. A 81, 025801 (2010).

V. E. Lobanov, Y. V. Kartashov, and L. Torner, “Light bullets by synthetic diffraction-dispersion matching,” Phys. Rev. Lett. 105, 033901 (2010).
[Crossref]

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802(R) (2010).

2009 (2)

I. B. Burgess, M. Peccianti, G. Assanto, and R. Morandotti, “Accessible light bullets via synergetic nonlinearities,” Phys. Rev. Lett. 102, 203903 (2009).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

2008 (3)

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (2008).

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT-symmetric potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

M. Belic, N. Petrovic, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

2007 (2)

2006 (4)

J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Optical bullets and rockets” in nonlineardissipative systems and their transformations and interactions,” Opt. Express 14, 4013–4025 (2006).
[Crossref]

V. Sharka and N. B. Aleksic, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96, 013903 (2006).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 025601 (2006).

2005 (7)

P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express 13, 9352–9360 (2005).
[Crossref]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, L. C. Crasovan, Y. V. Kartashov, and L. Torner, “Stable three-dimensional solitons in attractive Bose-Einstein condensates loaded in an optical lattice,” Phys. Rev. A 72, 021601 (2005).

T. J. Alexander, E. A. Ostrovskaya, A. A. Sukhorukov, and Y. S. Kivshar, “Three-dimensional matter-wave vortices in optical lattices,” Phys. Rev. A 72, 043603 (2005).

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).

M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30, 2140–2142 (2005).
[Crossref]

2004 (3)

P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, and R. Carretero-Gonzalez, “Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice,” Phys. Rev. Lett. 93, 080403 (2004).
[Crossref]

G. Fibich and B. Ilan, “Optical light bullets in a pure Kerr medium,” Opt. Lett. 29, 887–889 (2004).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E 70, 055603(R) (2004).

2003 (2)

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
[Crossref]

J. Yang and Z. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28, 2094–2096 (2003).
[Crossref]

2002 (3)

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[Crossref]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).

F. Wise and P. Di Trapani, “The hunt for light bullets,” Opt. Photon. News 13(2), 28 (2002).
[Crossref]

2001 (1)

L. Torner, S. Carrasco, J. P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
[Crossref]

2000 (4)

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljacic, “Composite multi hump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[Crossref]

S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. 180, 377–382 (2000).
[Crossref]

A. Desyatnikov, A. I. Maimistov, and B. A. Malomed, “Three-dimensional spinning solitons in dispersive media with cubic-quintic nonlinearity,” Phys. Rev. E 61, 3107–3113 (2000).

I. V. Melnikov, D. Mihalache, and N.-C. Panoiu, “Localized multidimensional femtosecond optical pulses in an off-resonance two-level medium,” Opt. Commun. 181, 345–351 (2000).
[Crossref]

1999 (1)

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4634 (1999).
[Crossref]

1998 (3)

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun. 148, 79–84 (1998).
[Crossref]

L. Berge, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
[Crossref]

N. A. Kaliteevskii, N. N. Rosanov, and S. V. Fedorov, “Formation of laser bullets,” Opt. Spectrosc. 85, 485–489 (1998).

1997 (1)

1995 (1)

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[Crossref]

1994 (1)

1992 (1)

1990 (1)

1985 (1)

S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 797–801 (1985).
[Crossref]

1981 (1)

A. A. Kanashov and A. M. Rubenchik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122–134 (1981).
[Crossref]

Abdullaev, F. K.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805(R) (2011).

Ablowitz, M. J.

Aceves, A. B.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[Crossref]

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).
[Crossref]

Agrawal, G. P.

S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. 180, 377–382 (2000).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Akhmediev, N.

Aleksic, N. B.

V. Sharka and N. B. Aleksic, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96, 013903 (2006).
[Crossref]

Alexander, T. J.

T. J. Alexander, E. A. Ostrovskaya, A. A. Sukhorukov, and Y. S. Kivshar, “Three-dimensional matter-wave vortices in optical lattices,” Phys. Rev. A 72, 043603 (2005).

Assanto, G.

I. B. Burgess, M. Peccianti, G. Assanto, and R. Morandotti, “Accessible light bullets via synergetic nonlinearities,” Phys. Rev. Lett. 102, 203903 (2009).
[Crossref]

Baizakov, B. B.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
[Crossref]

Bang, O.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).

Bartelt, H.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3, 041031 (2013).
[Crossref]

S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-dimensional light bullets in arrays of waveguides,” Phys. Rev. Lett. 105, 263901 (2010).
[Crossref]

Belic, M.

M. Belic, N. Petrovic, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

Bender, C. M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

Berge, L.

L. Berge, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
[Crossref]

Bersch, C.

C. Bersch, G. Onishchukov, and U. Peschel, “Optical gap solitons and truncated nonlinear Bloch waves in temporal lattices,” Phys. Rev. Lett. 109, 093903 (2012).
[Crossref]

Burgess, I. B.

I. B. Burgess, M. Peccianti, G. Assanto, and R. Morandotti, “Accessible light bullets via synergetic nonlinearities,” Phys. Rev. Lett. 102, 203903 (2009).
[Crossref]

Buryak, A. V.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[Crossref]

Carrasco, S.

L. Torner, S. Carrasco, J. P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
[Crossref]

Carretero-Gonzalez, R.

P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, and R. Carretero-Gonzalez, “Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice,” Phys. Rev. Lett. 93, 080403 (2004).
[Crossref]

Chen, G.

M. Belic, N. Petrovic, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

Chen, H.

J. Lien, Y. Chen, N. Ishida, H. Chen, C. Hwang, and F. Nori, “Multistability and condensation of exciton-polaritons below threshold,” Phys. Rev. B 91, 024511 (2014).

Chen, Y.

J. Lien, Y. Chen, N. Ishida, H. Chen, C. Hwang, and F. Nori, “Multistability and condensation of exciton-polaritons below threshold,” Phys. Rev. B 91, 024511 (2014).

Chen, Y.-X.

Y.-X. Chen and C.-Q. Dai, “Gaussian-type light bullets in power-law nonlinear media with PT-symmetric potential,” Opt. Commun. 338, 393–398 (2015).
[Crossref]

Chen, Z.

Christodoulides, D. N.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT-symmetric potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljacic, “Composite multi hump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[Crossref]

Cong, L.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT-symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Couairon, A.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durécu, A. Couairon, and A. Mysyrowicz, “Self-guided propagation of ultrashort laser pulses in the anomalous dispersion region of transparent solids: a new regime of filamentation,” Phys. Rev. Lett. 110, 115003 (2013).
[Crossref]

Crasovan, L. C.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, L. C. Crasovan, Y. V. Kartashov, and L. Torner, “Stable three-dimensional solitons in attractive Bose-Einstein condensates loaded in an optical lattice,” Phys. Rev. A 72, 021601 (2005).

Crasovan, L.-C.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 025601 (2006).

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E 70, 055603(R) (2004).

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[Crossref]

L. Torner, S. Carrasco, J. P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
[Crossref]

Dai, C.-Q.

Y.-X. Chen and C.-Q. Dai, “Gaussian-type light bullets in power-law nonlinear media with PT-symmetric potential,” Opt. Commun. 338, 393–398 (2015).
[Crossref]

C.-Q. Dai, X.-G. Wang, and G.-Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).

De Angelis, C.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[Crossref]

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994).
[Crossref]

De La Rue, R. M.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802(R) (2010).

de Nobriga, C. E.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802(R) (2010).

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).

Desyatnikov, A.

A. Desyatnikov, A. I. Maimistov, and B. A. Malomed, “Three-dimensional spinning solitons in dispersive media with cubic-quintic nonlinearity,” Phys. Rev. E 61, 3107–3113 (2000).

Di Trapani, P.

F. Wise and P. Di Trapani, “The hunt for light bullets,” Opt. Photon. News 13(2), 28 (2002).
[Crossref]

Ding, W.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802(R) (2010).

Dmitriev, S. V.

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, “Solitons in a chain of parity-time-invariant dimers,” Phys. Rev. E 84, 046609 (2011).

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35, 2976–2978 (2010).
[Crossref]

Dorofeenko, A. V.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Durand, M.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durécu, A. Couairon, and A. Mysyrowicz, “Self-guided propagation of ultrashort laser pulses in the anomalous dispersion region of transparent solids: a new regime of filamentation,” Phys. Rev. Lett. 110, 115003 (2013).
[Crossref]

Durécu, A.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durécu, A. Couairon, and A. Mysyrowicz, “Self-guided propagation of ultrashort laser pulses in the anomalous dispersion region of transparent solids: a new regime of filamentation,” Phys. Rev. Lett. 110, 115003 (2013).
[Crossref]

Edmundson, D. E.

Eilenberger, F.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3, 041031 (2013).
[Crossref]

S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-dimensional light bullets in arrays of waveguides,” Phys. Rev. Lett. 105, 263901 (2010).
[Crossref]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT-symmetric potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Enns, R. H.

Fan, S.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).

Fedorov, S. V.

N. A. Kaliteevskii, N. N. Rosanov, and S. V. Fedorov, “Formation of laser bullets,” Opt. Spectrosc. 85, 485–489 (1998).

Feng, L.

L. Feng, Z. J. Wong, R. Ma, Y. Wang, and X. Zhang, “Single-mode laser by PT-symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Fibich, G.

Firth, W. J.

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun. 148, 79–84 (1998).
[Crossref]

Forget, N.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durécu, A. Couairon, and A. Mysyrowicz, “Self-guided propagation of ultrashort laser pulses in the anomalous dispersion region of transparent solids: a new regime of filamentation,” Phys. Rev. Lett. 110, 115003 (2013).
[Crossref]

Frantzeskakis, D. J.

P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, and R. Carretero-Gonzalez, “Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice,” Phys. Rev. Lett. 93, 080403 (2004).
[Crossref]

Fu, S.

Garanovich, I. L.

M. Matuszewski, I. L. Garanovich, and A. A. Sukhorukov, “Light bullets in nonlinear periodically curved waveguide arrays,” Phys. Rev. A 81, 043833 (2010).

Ge, L.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).

Geiss, R.

F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and T. Pertsch, “Observation of discrete, vortex light bullets,” Phys. Rev. X 3, 041031 (2013).
[Crossref]

Gianfreda, M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).

Gorbach, A. V.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802(R) (2010).

Grabielle, S.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durécu, A. Couairon, and A. Mysyrowicz, “Self-guided propagation of ultrashort laser pulses in the anomalous dispersion region of transparent solids: a new regime of filamentation,” Phys. Rev. Lett. 110, 115003 (2013).
[Crossref]

Grelu, P.

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Han, J.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT-symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Hang, C.

C. Hang and G.-X. Huang, “Weak-light solitons and their active control in a parity-time-symmetric atomic system,” Phys. Rev. A 91, 043833 (2015).

C. Hang, G. Huang, and V. V. Konotop, “PT symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

He, Y.

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36, 2680–2682 (2011).
[Crossref]

Heinrich, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Hobbs, G. D.

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D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
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D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E 70, 055603(R) (2004).

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S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-dimensional light bullets in arrays of waveguides,” Phys. Rev. Lett. 105, 263901 (2010).
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D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
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D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E 70, 055603(R) (2004).

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
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D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 025601 (2006).

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
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D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
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D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 025601 (2006).

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
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D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
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D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable three-dimensional spatiotemporal solitons in a two-dimensional photonic lattice,” Phys. Rev. E 70, 055603(R) (2004).

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
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D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 025601 (2006).

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
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B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
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D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
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D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
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H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
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S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-dimensional light bullets in arrays of waveguides,” Phys. Rev. Lett. 105, 263901 (2010).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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Wang, H.

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
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Europhys. Lett. (1)

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
[Crossref]

J. Comput. Phys. (1)

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J. Opt. B (1)

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Figures (5)

Fig. 1.
Fig. 1. Bandgap spectrum of the two-dimensional PT -symmetric lattice below [panel (a), p i = 5 ] and above [panel (b), p i = 7 ] the PT -symmetry breaking point. Only the real part of the propagation constant in the first two bands is shown. (c) Evolution of maximal real b re max and maximal and minimal imaginary b im max , b im min parts of propagation constants of Bloch waves from the first two bands with an increase of p i . In all cases, p r = 6 .
Fig. 2.
Fig. 2. Dependencies of energy of a fundamental 3D soliton on b (a) for different p r values at p i = 2 and (b) for different p i values at p r = 6 . In (a), the depth of the real part of the lattice increases by 1 in the direction shown by the arrow. In (b), the curves are shown for p i = 1 , 5 , 7 and 9. (c) Peak amplitude a max of fundamental light bullet, its temporal width w τ , and the ratio of spatial and temporal widths w η w τ 1 as functions of b for p r = 6 , p i = 3 . (d) It is the same as in (a), but for the 3D vortex solitons.
Fig. 3.
Fig. 3. Profiles of stationary fundamental 3D solitons at b = 3.8 , p i = 1 (left column), b = 6 , p i = 1 (central column), and of l = + 1 vortex solitons at b = 0.5 , p i = 6 (right column). The first row shows isosurface field modulus distributions at the level | q | = 0.03 (left and central columns) and at the level | q | = 0.1 (right column). The second and third rows show field modulus and phase distributions at τ = 0 , the fourth row shows | q | at η = 0 (except for the right column, where the fourth row shows | q | τ = 0 for l = 1 soliton), and the fifth row shows phase ψ at η = 0 . Solitons in the left and central columns correspond to the red dots in Fig. 2(b). In all cases, p r = 6 . Distributions in the second–fifth rows are shown in the η , ζ , τ [ 4 , + 4 ] windows.
Fig. 4.
Fig. 4. Profile of a stable l = + 1 vortex soliton with b = 4.4 , p i = 0.55 . Isosurface depicting the field modulus distribution (left) at | q | = 0.07 .
Fig. 5.
Fig. 5. Domains of stability on the plane ( p i , b ) for fundamental (a) and vortex (b) 3D solitons at p r = 6 . Solitons are stable in the region between the red dots at b cr low < b < b cr upp . The black dots denote the cutoff for the fundamental 3D solitons. (c) Real part of perturbation growth rate versus b for vortex 3D solutions for p i = 0.3 , corresponding to the vertical dashed line in panel (b).

Equations (5)

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i q ξ = 1 2 ( 2 q η 2 + 2 q ζ 2 ) β 2 2 q τ 2 q | q | 2 R ( η , ζ ) q .
b m a m = 1 2 ( 2 η 2 + 2 ζ 2 + 2 i k η η + 2 i k ζ ζ k η 2 k ζ 2 ) a m + R ( η , ζ ) a m ,
b | a | = ( 1 / 2 ) 2 | a | j 2 / ( 2 | a | 3 ) + | a | R r + | a | 3 , · j = 2 | a | 2 R i ,
w η 2 = U 1 η 2 | a | 2 d η d ζ d τ , w τ 2 = U 1 τ 2 | a | 2 d η d ζ d τ
L = i ( A 2 + i R i Δ 2 + 2 | a | 2 A + 2 + R r b Δ 2 + 2 | a | 2 + A + 2 + R r b A 2 + i R i ) ,

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