Abstract

We consider the model of a dual-core spatial-domain coupler with χ(2) and χ(3) nonlinearities acting in two parallel cores. We construct families of symmetric and asymmetric solitons in the system with self-defocusing χ(3) terms and test their stability. The transition from symmetric to asymmetric soliton branches and back to the symmetric ones proceeds via a bifurcation loop. Namely, a pair of stable asymmetric branches emerges from the symmetric family via a supercritical bifurcation; eventually, the asymmetric branches merge back into the symmetric one through a reverse bifurcation. The existence of the loop is explained by means of an extended version of the cascading approximation for the χ(2) interaction, which takes into regard the cross-phase modulation part of the χ(3) interaction. When the intercore coupling is weak, the bifurcation loop features a concave shape, with the asymmetric branches losing their stability at the turning points. In addition to the two-color solitons, which are built of the fundamental-frequency (FF) and second-harmonic (SH) components, in the case of the self-focusing χ(3) nonlinearity we also consider single-color solitons, which contain only the SH component but may be subject to the instability against FF perturbations. Asymmetric single-color solitons are always unstable, whereas the symmetric ones are stable, provided that they do not coexist with two-color counterparts. Collisions between tilted solitons are studied, too.

© 2013 Optical Society of America

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  5. A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
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  29. R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36, 4323–4325 (2011).
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    [CrossRef]
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  32. W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Asymmetric solitons in coupled second-harmonic-generating waveguides,” Phys. Rev. E 57, 1092–1103 (1998).
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    [CrossRef]
  36. R. Driben, B. A. Malomed, and P. L. Chu, “All-optical switching in a two-channel waveguide with cubic–quintic nonlinearity,” J. Phys. B 39, 2455–2466 (2006).
    [CrossRef]
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    [CrossRef]
  38. H. Hatami-Hanza, P. L. Chu, and G. D. Peng, “Optical switching in a coupler with saturable nonlinearity,” Opt. Quantum Electron. 26, S365–S372 (1994).
    [CrossRef]
  39. A. Ankiewicz, N. Akhmediev, and G. P. Peng, “Stationary soliton states in couplers with saturable nonlinearity,” Opt. Quantum Electron. 27, 193–200 (1995).
    [CrossRef]
  40. A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Novel bifurcation phenomena for solitons in nonlinear saturable couplers,” Opt. Commun. 116, 411–415 (1995).
    [CrossRef]
  41. R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
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2012 (1)

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[CrossRef]

2011 (1)

2008 (2)

Z. Birnbaum and B. A. Malomed, “Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity,” Physica D 237, 3252–3262 (2008).
[CrossRef]

M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008).
[CrossRef]

2007 (2)

T. Ellenbogen, A. Arie, and M. Solomon, “Noncollinear double quasi phase matching in one-dimensional poled crystal,” Opt. Lett. 32, 262–264 (2007).
[CrossRef]

L. Albuch and B. A. Malomed, “Transitions between symmetric and asymmetric solitons in dual-core systems with cubic-quintic nonlinearity,” Math. Comput. Simul. 74, 312–322 (2007).

2006 (1)

R. Driben, B. A. Malomed, and P. L. Chu, “All-optical switching in a two-channel waveguide with cubic–quintic nonlinearity,” J. Phys. B 39, 2455–2466 (2006).
[CrossRef]

2003 (1)

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

2002 (1)

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

2001 (3)

J. F. Corney and O. Bang, “Modulational instability in periodic quadratic nonlinear materials,” Phys. Rev. Lett. 87, 133901 (2001).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

2000 (1)

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

1999 (1)

1998 (2)

O. Bang, Yu. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearity,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Asymmetric solitons in coupled second-harmonic-generating waveguides,” Phys. Rev. E 57, 1092–1103 (1998).
[CrossRef]

1997 (5)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Solitons in coupled waveguides with quadratic nonlinearity,” Phys. Rev. E 55, 6134–6140 (1997).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

O. Bang, Yu. S. Kivshar, and A. V. Buryak, “Bright spatial solitons in defocusing Kerr media supported by cascaded nonlinearities,” Opt. Lett. 22, 1680–1682 (1997).
[CrossRef]

O. Bang, “Dynamical equations for wave packets in materials with both quadratic and cubic response,” J. Opt. Soc. Am. B 14, 51–61 (1997).
[CrossRef]

1996 (2)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

B. A. Malomed, I. Skinner, P. L. Chu, and G. D. Peng, “Symmetric and asymmetric solitons in twin-core nonlinear optical fibers,” Phys. Rev. E 53, 4084–4091 (1996).
[CrossRef]

1995 (3)

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett. 20, 1961–1963 (1995).
[CrossRef]

A. Ankiewicz, N. Akhmediev, and G. P. Peng, “Stationary soliton states in couplers with saturable nonlinearity,” Opt. Quantum Electron. 27, 193–200 (1995).
[CrossRef]

A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Novel bifurcation phenomena for solitons in nonlinear saturable couplers,” Opt. Commun. 116, 411–415 (1995).
[CrossRef]

1994 (3)

G. D. Peng, P. L. Chu, and A. Ankiewicz, “Soliton propagation in saturable nonlinear fiber couplers—variational and numerical results,” Int. J. Nonlinear Opt. Phys. 3, 69–87 (1994).
[CrossRef]

H. Hatami-Hanza, P. L. Chu, and G. D. Peng, “Optical switching in a coupler with saturable nonlinearity,” Opt. Quantum Electron. 26, S365–S372 (1994).
[CrossRef]

H. Lee, W. R. Cho, J. H. Park, J. S. Kim, S. H. Park, and U. Kim, “Measurement of free-carrier nonlinearities in ZnSe based on the Z-scan technique with a nanosecond laser,” Opt. Lett. 19, 1116–1118 (1994).
[CrossRef]

1993 (3)

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef]

P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analytical results,” J. Opt. Soc. Am. B 10, 1379–1385 (1993).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

1992 (3)

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68, 3547–3550 (1992).
[CrossRef]

M. V. Komissarova and A. P. Sukhorukov, “Optical solitons in media with quadratic and cubic nonlinearities,” Bull. Russ. Acad. Sci. Phys. 56, 1995–1999 (1992).

1991 (1)

A. I. Maimistov, “Propagation of a light pulse in nonlinear tunnel-coupled optical waveguides,” Kvant. Elektron. 18, 758–761 (1991) [Sov. J. Quantum Electron. 21, 687–690 (1991)].

1990 (1)

C. Paré and M. Fłorjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[CrossRef]

1989 (1)

E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers,” Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef]

1961 (1)

P. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. E 7, 118–120 (1961).

Akhmediev, N.

A. Ankiewicz, N. Akhmediev, and G. P. Peng, “Stationary soliton states in couplers with saturable nonlinearity,” Opt. Quantum Electron. 27, 193–200 (1995).
[CrossRef]

A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Novel bifurcation phenomena for solitons in nonlinear saturable couplers,” Opt. Commun. 116, 411–415 (1995).
[CrossRef]

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

Albuch, L.

L. Albuch and B. A. Malomed, “Transitions between symmetric and asymmetric solitons in dual-core systems with cubic-quintic nonlinearity,” Math. Comput. Simul. 74, 312–322 (2007).

Alexeeva, N. V.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[CrossRef]

Ankiewicz, A.

A. Ankiewicz, N. Akhmediev, and G. P. Peng, “Stationary soliton states in couplers with saturable nonlinearity,” Opt. Quantum Electron. 27, 193–200 (1995).
[CrossRef]

A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Novel bifurcation phenomena for solitons in nonlinear saturable couplers,” Opt. Commun. 116, 411–415 (1995).
[CrossRef]

G. D. Peng, P. L. Chu, and A. Ankiewicz, “Soliton propagation in saturable nonlinear fiber couplers—variational and numerical results,” Int. J. Nonlinear Opt. Phys. 3, 69–87 (1994).
[CrossRef]

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef]

Arie, A.

Bache, M.

Bang, O.

M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008).
[CrossRef]

J. F. Corney and O. Bang, “Modulational instability in periodic quadratic nonlinear materials,” Phys. Rev. Lett. 87, 133901 (2001).
[CrossRef]

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999).
[CrossRef]

O. Bang, Yu. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearity,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

O. Bang, Yu. S. Kivshar, and A. V. Buryak, “Bright spatial solitons in defocusing Kerr media supported by cascaded nonlinearities,” Opt. Lett. 22, 1680–1682 (1997).
[CrossRef]

O. Bang, “Dynamical equations for wave packets in materials with both quadratic and cubic response,” J. Opt. Soc. Am. B 14, 51–61 (1997).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

Barashenkov, I. V.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[CrossRef]

Birnbaum, Z.

Z. Birnbaum and B. A. Malomed, “Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity,” Physica D 237, 3252–3262 (2008).
[CrossRef]

Bramati, A.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Brambilla, M.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

Buryak, A. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

O. Bang, Yu. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearity,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

O. Bang, Yu. S. Kivshar, and A. V. Buryak, “Bright spatial solitons in defocusing Kerr media supported by cascaded nonlinearities,” Opt. Lett. 22, 1680–1682 (1997).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett. 20, 1961–1963 (1995).
[CrossRef]

Chinaglia, W.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Cho, W. R.

Christiansen, P. L.

Chu, P. L.

R. Driben, B. A. Malomed, and P. L. Chu, “All-optical switching in a two-channel waveguide with cubic–quintic nonlinearity,” J. Phys. B 39, 2455–2466 (2006).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Asymmetric solitons in coupled second-harmonic-generating waveguides,” Phys. Rev. E 57, 1092–1103 (1998).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Solitons in coupled waveguides with quadratic nonlinearity,” Phys. Rev. E 55, 6134–6140 (1997).
[CrossRef]

B. A. Malomed, I. Skinner, P. L. Chu, and G. D. Peng, “Symmetric and asymmetric solitons in twin-core nonlinear optical fibers,” Phys. Rev. E 53, 4084–4091 (1996).
[CrossRef]

G. D. Peng, P. L. Chu, and A. Ankiewicz, “Soliton propagation in saturable nonlinear fiber couplers—variational and numerical results,” Int. J. Nonlinear Opt. Phys. 3, 69–87 (1994).
[CrossRef]

H. Hatami-Hanza, P. L. Chu, and G. D. Peng, “Optical switching in a coupler with saturable nonlinearity,” Opt. Quantum Electron. 26, S365–S372 (1994).
[CrossRef]

P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analytical results,” J. Opt. Soc. Am. B 10, 1379–1385 (1993).
[CrossRef]

Clausen, C. B.

O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

Conti, C.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Corney, J. F.

J. F. Corney and O. Bang, “Modulational instability in periodic quadratic nonlinear materials,” Phys. Rev. Lett. 87, 133901 (2001).
[CrossRef]

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

de Araújo, C. B.

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68, 3547–3550 (1992).
[CrossRef]

De Rossi, A.

O. Bang, Yu. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearity,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

Di Trapani, P.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Driben, R.

R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36, 4323–4325 (2011).
[CrossRef]

R. Driben, B. A. Malomed, and P. L. Chu, “All-optical switching in a two-channel waveguide with cubic–quintic nonlinearity,” J. Phys. B 39, 2455–2466 (2006).
[CrossRef]

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

Ellenbogen, T.

Etrich, C.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

Firth, W. J.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

Florjanczyk, M.

C. Paré and M. Fłorjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[CrossRef]

Franken, P.

P. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. E 7, 118–120 (1961).

Gomes, A. S. L.

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68, 3547–3550 (1992).
[CrossRef]

Gutin, M.

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Hatami-Hanza, H.

H. Hatami-Hanza, P. L. Chu, and G. D. Peng, “Optical switching in a coupler with saturable nonlinearity,” Opt. Quantum Electron. 26, S365–S372 (1994).
[CrossRef]

Hickmann, J. M.

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68, 3547–3550 (1992).
[CrossRef]

Hill, A. E.

P. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. E 7, 118–120 (1961).

Iooss, G.

G. Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory (Springer, 1980).

Joseph, D. D.

G. Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory (Springer, 1980).

Kilius, J.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Kim, J. S.

Kim, U.

Kivshar, Y. S.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

Kivshar, Yu. S.

Komissarova, M. V.

M. V. Komissarova and A. P. Sukhorukov, “Optical solitons in media with quadratic and cubic nonlinearities,” Bull. Russ. Acad. Sci. Phys. 56, 1995–1999 (1992).

Królikowski, W.

Lederer, F.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

Lee, H.

Lugiato, L. A.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

Mahlab, U.

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

Maimistov, A. I.

A. I. Maimistov, “Propagation of a light pulse in nonlinear tunnel-coupled optical waveguides,” Kvant. Elektron. 18, 758–761 (1991) [Sov. J. Quantum Electron. 21, 687–690 (1991)].

Mak, W. C. K.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Asymmetric solitons in coupled second-harmonic-generating waveguides,” Phys. Rev. E 57, 1092–1103 (1998).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Solitons in coupled waveguides with quadratic nonlinearity,” Phys. Rev. E 55, 6134–6140 (1997).
[CrossRef]

Malomed, B. A.

R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36, 4323–4325 (2011).
[CrossRef]

Z. Birnbaum and B. A. Malomed, “Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity,” Physica D 237, 3252–3262 (2008).
[CrossRef]

L. Albuch and B. A. Malomed, “Transitions between symmetric and asymmetric solitons in dual-core systems with cubic-quintic nonlinearity,” Math. Comput. Simul. 74, 312–322 (2007).

R. Driben, B. A. Malomed, and P. L. Chu, “All-optical switching in a two-channel waveguide with cubic–quintic nonlinearity,” J. Phys. B 39, 2455–2466 (2006).
[CrossRef]

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Asymmetric solitons in coupled second-harmonic-generating waveguides,” Phys. Rev. E 57, 1092–1103 (1998).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Solitons in coupled waveguides with quadratic nonlinearity,” Phys. Rev. E 55, 6134–6140 (1997).
[CrossRef]

B. A. Malomed, I. Skinner, P. L. Chu, and G. D. Peng, “Symmetric and asymmetric solitons in twin-core nonlinear optical fibers,” Phys. Rev. E 53, 4084–4091 (1996).
[CrossRef]

P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analytical results,” J. Opt. Soc. Am. B 10, 1379–1385 (1993).
[CrossRef]

Minardi, S.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Moses, J.

New, G.

G. New, Introduction to Nonlinear Optics (Cambridge University, 2011).

Paré, C.

C. Paré and M. Fłorjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[CrossRef]

Park, J. H.

Park, S. H.

Peng, G. D.

B. A. Malomed, I. Skinner, P. L. Chu, and G. D. Peng, “Symmetric and asymmetric solitons in twin-core nonlinear optical fibers,” Phys. Rev. E 53, 4084–4091 (1996).
[CrossRef]

H. Hatami-Hanza, P. L. Chu, and G. D. Peng, “Optical switching in a coupler with saturable nonlinearity,” Opt. Quantum Electron. 26, S365–S372 (1994).
[CrossRef]

G. D. Peng, P. L. Chu, and A. Ankiewicz, “Soliton propagation in saturable nonlinear fiber couplers—variational and numerical results,” Int. J. Nonlinear Opt. Phys. 3, 69–87 (1994).
[CrossRef]

P. L. Chu, B. A. Malomed, and G. D. Peng, “Soliton switching and propagation in nonlinear fiber couplers: analytical results,” J. Opt. Soc. Am. B 10, 1379–1385 (1993).
[CrossRef]

Peng, G. P.

A. Ankiewicz, N. Akhmediev, and G. P. Peng, “Stationary soliton states in couplers with saturable nonlinearity,” Opt. Quantum Electron. 27, 193–200 (1995).
[CrossRef]

Peschel, T.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

Peschel, U.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

Peters, C. W.

P. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. E 7, 118–120 (1961).

Prati, F.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

Romagnoli, M.

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

Skinner, I.

B. A. Malomed, I. Skinner, P. L. Chu, and G. D. Peng, “Symmetric and asymmetric solitons in twin-core nonlinear optical fibers,” Phys. Rev. E 53, 4084–4091 (1996).
[CrossRef]

Skryabin, D. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Solomon, M.

Soto-Crespo, J. M.

A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Novel bifurcation phenomena for solitons in nonlinear saturable couplers,” Opt. Commun. 116, 411–415 (1995).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

Spinelli, L.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers,” Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef]

Sukhorukov, A. A.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[CrossRef]

Sukhorukov, A. P.

M. V. Komissarova and A. P. Sukhorukov, “Optical solitons in media with quadratic and cubic nonlinearities,” Bull. Russ. Acad. Sci. Phys. 56, 1995–1999 (1992).

Torner, L.

O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Trillo, S.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

O. Bang, Yu. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearity,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett. 20, 1961–1963 (1995).
[CrossRef]

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

Valiulis, G.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Wabnitz, S.

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers,” Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef]

Weinreich, G.

P. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. E 7, 118–120 (1961).

Wise, F. W.

Wright, E. M.

E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers,” Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef]

Bull. Russ. Acad. Sci. Phys. (1)

M. V. Komissarova and A. P. Sukhorukov, “Optical solitons in media with quadratic and cubic nonlinearities,” Bull. Russ. Acad. Sci. Phys. 56, 1995–1999 (1992).

Int. J. Nonlinear Opt. Phys. (1)

G. D. Peng, P. L. Chu, and A. Ankiewicz, “Soliton propagation in saturable nonlinear fiber couplers—variational and numerical results,” Int. J. Nonlinear Opt. Phys. 3, 69–87 (1994).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. B (1)

R. Driben, B. A. Malomed, and P. L. Chu, “All-optical switching in a two-channel waveguide with cubic–quintic nonlinearity,” J. Phys. B 39, 2455–2466 (2006).
[CrossRef]

Kvant. Elektron. (1)

A. I. Maimistov, “Propagation of a light pulse in nonlinear tunnel-coupled optical waveguides,” Kvant. Elektron. 18, 758–761 (1991) [Sov. J. Quantum Electron. 21, 687–690 (1991)].

Math. Comput. Simul. (1)

L. Albuch and B. A. Malomed, “Transitions between symmetric and asymmetric solitons in dual-core systems with cubic-quintic nonlinearity,” Math. Comput. Simul. 74, 312–322 (2007).

Opt. Commun. (2)

A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Novel bifurcation phenomena for solitons in nonlinear saturable couplers,” Opt. Commun. 116, 411–415 (1995).
[CrossRef]

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Opt. Quantum Electron. (4)

H. Hatami-Hanza, P. L. Chu, and G. D. Peng, “Optical switching in a coupler with saturable nonlinearity,” Opt. Quantum Electron. 26, S365–S372 (1994).
[CrossRef]

A. Ankiewicz, N. Akhmediev, and G. P. Peng, “Stationary soliton states in couplers with saturable nonlinearity,” Opt. Quantum Electron. 27, 193–200 (1995).
[CrossRef]

M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Phys. Rep. (1)

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Phys. Rev. A (3)

E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers,” Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef]

C. Paré and M. Fłorjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[CrossRef]

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[CrossRef]

Phys. Rev. E (7)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Solitons in coupled waveguides with quadratic nonlinearity,” Phys. Rev. E 55, 6134–6140 (1997).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Asymmetric solitons in coupled second-harmonic-generating waveguides,” Phys. Rev. E 57, 1092–1103 (1998).
[CrossRef]

O. Bang, Yu. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearity,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Stability of the soliton states in a nonlinear fiber coupler,” Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

B. A. Malomed, I. Skinner, P. L. Chu, and G. D. Peng, “Symmetric and asymmetric solitons in twin-core nonlinear optical fibers,” Phys. Rev. E 53, 4084–4091 (1996).
[CrossRef]

P. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. E 7, 118–120 (1961).

Phys. Rev. Lett. (6)

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

J. F. Corney and O. Bang, “Modulational instability in periodic quadratic nonlinear materials,” Phys. Rev. Lett. 87, 133901 (2001).
[CrossRef]

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68, 3547–3550 (1992).
[CrossRef]

N. Akhmediev and A. Ankiewicz, “Novel soliton states and bifurcation phenomena in nonlinear fiber couplers,” Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef]

Physica D (1)

Z. Birnbaum and B. A. Malomed, “Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity,” Physica D 237, 3252–3262 (2008).
[CrossRef]

Prog. Opt. (1)

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with a quadratic nonlinearity,” Prog. Opt. 41, 483–568 (2000).
[CrossRef]

Other (3)

G. New, Introduction to Nonlinear Optics (Cambridge University, 2011).

B. A Malomed, ed., Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations (Springer, 2013).

G. Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory (Springer, 1980).

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Figures (13)

Fig. 1.
Fig. 1.

Concave bifurcation loops for solitons, obtained at different values of the FF intercore coupling coefficient Q, are displayed by means of the dependence of asymmetry measure (5) on the total power (norm), P: (a) Q=0.0047, (b) Q=0.0054, (c) Q=0.0065, and (d) Q=0.0085. Here and in Fig. 2, as well in Fig. 11 below, stable and unstable portions of the solution branches are shown by solid and dashed–dotted segments, respectively. The dot on the unstable asymmetric branch indicates a solution whose spontaneous transformation into a symmetric breather is displayed below in Fig. 4. This figure and those following below are drawn for α=0.5.

Fig. 2.
Fig. 2.

Continuation of Fig. (1). Convex bifurcation loops obtained at larger values of the FF coupling constant: (a) Q=0.0098, (b) Q=0.0106, (c) Q=0.0115, and (d) Q=0.0117. In (c) and (d), the short unstable segment of the symmetric solitons inside the loops is not plotted.

Fig. 3.
Fig. 3.

Solid curve: numerically found critical value of the FF coupling constant, Qcr, at which the bifurcation loop collapses, versus the phase mismatch, α. The dashed–dotted curve represents the crude analytical approximation for Qcr given by Eq. (10).

Fig. 4.
Fig. 4.

Evolution, initiated by small perturbations, of the FF and SH components in cores 1 and 2 (left and right panels, respectively) of the unstable asymmetric soliton marked by the dot in Fig. 1(a), with Q=0.0047, propagation constant λ=0.0255, and total power P=15.77. In each panel, the plots of the FF and SH components are juxtaposed with a shift, for the purpose of clearer presentation.

Fig. 5.
Fig. 5.

Periodic out-of-phase oscillations of amplitudes of the FF fields in the two cores, corresponding to the situation displayed in Fig. 4.

Fig. 6.
Fig. 6.

Elastic collision between low-power solitons, for Q=0.003 and λ=0.0046 (which corresponds to total power P=0.5162). In this figure and similar ones which display collisions, the left- and right-hand plots represent, respectively, the FF fields in cores 1 and 2. The picture of the SH component is quite similar.

Fig. 7.
Fig. 7.

Inelastic collision between identical high-power asymmetric solitons, at Q=0.003 and λ=0.0248 (which corresponds to total power P=10.43).

Fig. 8.
Fig. 8.

Inelastic collision between strongly asymmetric solitons, one being a mirror image of the other (i.e., the larger component of one soliton collides with the smaller component of the other), for Q=0.003 and λ=0.0245, which corresponds to total power P=5.63.

Fig. 9.
Fig. 9.

Inelastic collision between identical symmetric solitons whose power exceeds the largest power of asymmetric ones; i.e., they are located to the right of the bifurcation loop, in terms of Figs. 1 and 2. The parameters are Q=0.003, λ=0.0267, and P=10.99.

Fig. 10.
Fig. 10.

Transition from two-color solitons in (a), (b) and (c), (d), with the dominant FF or SH components, respectively, to the single-color (SH) soliton in (e), (f), in the system with the self-focusing χ(3) nonlinearity (σ=+1), relatively strong intercore coupling in the SH field (Q=0.2, K=0.5), and mismatch α=1.3. The propagation constants of the solitons and powers of their components are λ=0.1,(Pu)1=1.5688, (Pw)1=0.3328, (Pu)2=0.0523, (Pw)2=0.0956 in (a), (b); λ=0.531, (Pu)1=0.2027, (Pw)1=1.8684, (Pu)2=0.1413, (Pw)2=1.4836 in (c), (d); and λ=0.494, (Pu)1,2=0, (Pw)1,2=1.678 in (e), (f). The two-color asymmetric soliton is stable in (a), (b) and unstable in (c), (d). The single-color symmetric soliton in (e), (f) is located at a boundary between stable and unstable subfamilies.

Fig. 11.
Fig. 11.

SBB of the single-color (SH) solitons. Asymmetry ΘS is defined as per Eq. (12). The bifurcation takes place at P4.47.

Fig. 12.
Fig. 12.

Evolution of an unstable asymmetric single-color soliton at α=2.32, with λ=0.8 and powers in the two cores (Pw)1=3.0056, (Pw)2=1.5616. The left- and right-hand panels represent the fields in cores 1 and core 2, respectively. In each panel, the FF and SH components are shifted to the left and right for clearer presentation.

Fig. 13.
Fig. 13.

SH single-component symmetric solitons are stable with the power taking values below the maximum value shown here, P<Pmax, which depends on the phase mismatch, α.

Equations (12)

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iu1z+u1xx+u1*w1+σ(14|u1|2+2|w1|2)u1+Qu2=0,2iw1z+w1xxαw1+12u12+σ(4|w1|2+2|u1|2)w1+Kw2=0,iu2z+u2xx+u2*w2+σ(14|u2|2+2|w2|2)u2+Qu1=0,2iw2z+w2xxαw2+12u22+σ(4|w2|2+2|u2|2)w2+Kw1=0.
P1,2=+[|u1,2(x)|2+4|w1,2(x)|2]dx,
{u1(x,z),w1(x,z)}={U1(x),W1(x)}eiλz,{u2(x,z),w2(x,z)}={U2(x),W2(x)}e2iλz,
λU1+U1+U1*W1+σ(14|U1|2+2|W1|2)U1+QU2=0,(4λ+α)W1+W1+12U12+σ(4|W1|2+2|U1|2)W1+KW2=0,λU2+U2+U2*W2+σ(14|U2|2+2|W2|2)U2+QU1=0,(4λ+α)W2+W2+12U22+σ(4|W2|2+2|U2|2)W2+KW1=0,
ΘF=(U1)max2(U2)max2(U1)max2+(U2)max2.
Wn(x)(Un(x))22[α+2|Un(x)|2],n=1,2,
λU1+U1xx+|U1|22(α+2|U1|2)U114|U1|2U1+QU2=0,λU2+U2xx+|U2|22(α+2|U2|2)U214|U2|2U2+QU1=0.
(χ1,2(3))eff=12[1α+2(U1,2)max212],
Pmax2+(16Q31+α2)Pmax+83αQ=0.
Qcr=332(2α)2.
u1,2(x,z)u1,2(xcz,z)exp(i2cxi4c2z),w1,2(x,z)w1,2(xcz,z)exp(icxi2c2z),
ΘS=(W1)max2(W2)max2(W1)max2+(W2)max2,

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