Classification of the solitary waves in coupled nonlinear Schrödinger equations. (English) Zbl 0938.35180

Summary: The solitary waves in coupled nonlinear Schrödinger equations are classified into infinite families. For each of the first three families, the parameter region is specified and the parameter dependence of its solitary waves is described and explained. We find that the parameter regions of these solution families are novel and irregular, and the parameter dependence of the solitary waves is sensitive. The stability of these families of solitary waves is also determined. We show that only the family of symmetric and single-humped solitary waves is stable.


35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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[1] Benney, D.J.; Newell, A.C., The propagation of nonlinear wave envelopes, J. math. and phys., 46, 133, (1967) · Zbl 0153.30301
[2] Roskes, G.J., Some nonlinear multiphase interactions, Stud. appl. math., 55, 231, (1976) · Zbl 0345.76012
[3] Menyuk, C.R., Nonlinear pulse propagation in birefringent optical fibers, IEEE J. quantum electron, QE-23, 174, (1987)
[4] Agrawal, G.P., Nonlinear fiber optics, (1995), Academic Press New York
[5] Yang, J.; Benney, D.J., Some properties of nonlinear wave systems, Stud. appl. math., 96, 111, (1996) · Zbl 0857.35117
[6] Yang, J., Vector solitons and their internal oscillations in birefringent nonlinear optical fibers, Stud. appl. math., 98, 61, (1997) · Zbl 0870.35111
[7] Yang, J., Multiple permanent-wave trains in nonlinear systems, Stud. appl. math., (1997), to appear
[8] Mesentsev, V.K.; Turitsyn, S.K., Stability of vector solitons in optical fibers, Opt. lett., 17, 1497, (1992)
[9] Kaup, D.J.; Malomed, B.A.; Tasgal, R.S., Internal dynamics of a vector soliton in a nonlinear optical fiber, Phys. rev. E, 48, 3049, (1993)
[10] Haelterman, M.; Sheppard, A.P., The elliptically polarized fundamental vector soliton of isotropic Kerr media, Phys. lett. A, 194, 191, (1994)
[11] Haelterman, M.; Sheppard, A.P., Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media, Phys. rev. E, 49, 3376, (1994)
[12] Haelterman, M.; Sheppard, A.P.; Snyder, A.W., Bound-vector solitary waves in isotropic nonlinear dispersive media, Opt. lett., 18, 1406, (1993)
[13] Silberberg, Y.; Barad, Y., Rotating vector solitary waves in isotropic fibers, Opt. lett., 20, 246, (1995)
[14] Balmforth, N.J., Solitary waves and homoclinic orbits, Ann. rev. fluid mech., 27, 335, (1995)
[15] Akhmediev, N.N.; Buryak, A.V.; Soto-Crespo, J.M.; Andersen, D.R., Phase-locked stationary soliton states in birefringent nonlinear optical fibers, J. opt. soc. am. B, 12, 434, (1995)
[16] Buffoni, B.; Sere, E., A global condition for quasi-random behavior in a class of conservative systems, Comm. pure appl. math., 49, 285, (1996) · Zbl 0860.58027
[17] Kalies, W.D.; VanderVorst, R.C.A.M., Multitransition homoclinic and heteroclinic solutions of the extended fisher—kolmogorov equation, CDSNS report # 231, (1995), Georgia Institute of Technology · Zbl 0872.34033
[18] P.H. Rabinowitz, Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. Var. 1: 1-36. · Zbl 0791.34042
[19] Landau, L.D.; Lifshitz, E.M., Quantum mechanics: non-relativistic theory, (1977), Pergamon Press Oxford · Zbl 0178.57901
[20] Kaup, D.J.; Lakoba, T.I.; Malomed, B.A., Asymmetric solitons in mismatched dual-core optical fibers, INS report # 271, (1996), Clarkson University
[21] Manakov, S.V., On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. phys. JETP, 38, 248-253, (1974)
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