Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions

The inverse scattering transform for the focusing nonlinear Schrodinger equation with non-zero boundary conditions at infinity is presented, including the determination of the analyticity of the scattering eigenfunctions, the introduction of the appropriate Riemann surface and uniformization variable, the symmetries, discrete spectrum, asymptotics, trace formulae and the so-called theta condition, and the formulation of the inverse problem in terms of a Riemann-Hilbert problem. In addition, the general behavior of the soliton solutions is discussed, as well as the reductions to all special cases previously discussed in the literature.

[1]  Federica Vitale,et al.  The Inverse Scattering Transform for the Defocusing Nonlinear Schrödinger Equations with Nonzero Boundary Conditions , 2013 .

[2]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1992, math/9201261.

[3]  N. Akhmediev,et al.  Waves that appear from nowhere and disappear without a trace , 2009 .

[4]  J. Soto-Crespo,et al.  Rogue waves and rational solutions of the nonlinear Schrödinger equation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Yan‐Chow Ma,et al.  The Perturbed Plane‐Wave Solutions of the Cubic Schrödinger Equation , 1979 .

[6]  J. Garnier,et al.  Inverse scattering perturbation theory for the nonlinear Schrödinger equation with non-vanishing background , 2012 .

[7]  I. Gabitov,et al.  Maxwell-Bloch equation and the inverse scattering method , 1985 .

[8]  J. K. Shaw,et al.  On the Eigenvalues of Zakharov-Shabat Systems , 2003, SIAM J. Math. Anal..

[9]  R. Dodd,et al.  Review: L. D. Faddeev and L. A. Takhtajan, Hamiltonian methods in the theory of solitons , 1988 .

[10]  P. Bélanger,et al.  Bright solitons on a cw background , 1996 .

[11]  J. Satsuma,et al.  B Initial Value Problems of One-Dimensional self-Modulation of Nonlinear Waves in Dispersive Media (Part V. Initial Value Problems) , 1975 .

[12]  L. Ostrovsky,et al.  Modulation instability: The beginning , 2009 .

[13]  G Ruocco,et al.  Free-energy transition in a gas of noninteracting nonlinear wave particles. , 2008, Physical review letters.

[14]  E. Kuznetsov,et al.  Solitons in a parametrically unstable plasma , 1977 .

[15]  T. Brooke Benjamin,et al.  The disintegration of wave trains on deep water Part 1. Theory , 1967, Journal of Fluid Mechanics.

[16]  Thomas Brooke Benjamin,et al.  Instability of periodic wavetrains in nonlinear dispersive systems , 1967, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[17]  Ronald R. Coifman,et al.  Scattering and inverse scattering for first order systems , 1984 .

[18]  P. Deift,et al.  The collisionless shock region for the long-time behavior of solutions of the KdV equation , 1994 .

[19]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[20]  D. H. Peregrine,et al.  Water waves, nonlinear Schrödinger equations and their solutions , 1983, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[21]  Athanassios S. Fokas,et al.  Complex Variables: Contents , 2003 .

[22]  Gino Biondini,et al.  Inverse Scattering Transform for the Multi‐Component Nonlinear Schrödinger Equation with Nonzero Boundary Conditions , 2011 .

[23]  J. K. Shaw,et al.  Purely imaginary eigenvalues of Zakharov-Shabat systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Mark J. Ablowitz,et al.  Coherent pulse propagation, a dispersive, irreversible phenomenon , 1974 .

[25]  Yosuke Watanabe,et al.  Breather solutions to the focusing nonlinear Schrödinger equation , 1998 .

[26]  B. Jalali,et al.  Optical rogue waves , 2007, Nature.

[27]  Gino Biondini,et al.  On the Spectrum of the Dirac Operator and the Existence of Discrete Eigenvalues for the Defocusing Nonlinear Schrödinger Equation , 2014 .

[28]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[29]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1993 .

[30]  V E Zakharov,et al.  Nonlinear stage of modulation instability. , 2012, Physical review letters.

[31]  A. A. Gelash,et al.  Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability , 2012, 1211.1426.

[32]  A. B. Shabat,et al.  Interaction between solitons in a stable medium , 1973 .

[33]  V. Zakharov,et al.  Soliton on Unstable Condensate , 2011, 1109.0620.

[34]  D. Mihalache,et al.  Two-parameter family of exact solutions of the nonlinear Schrödinger equation describing optical-soliton propagation. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[35]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[36]  A. Osborne,et al.  Modulational instability in crossing sea states: a possible mechanism for the formation of freak waves. , 2006, Physical review letters.

[37]  M. Gregory Forest,et al.  Geometry and Modulation Theory for the Periodic Nonlinear Schrodinger Equation , 1986 .

[38]  Mark J. Ablowitz,et al.  On homoclinic structure and numerically induced chaos for the nonlinear Schro¨dinger equation , 1990 .

[39]  D. J. Benney,et al.  The Propagation of Nonlinear Wave Envelopes , 1967 .

[40]  J. Fatome,et al.  Observation of Kuznetsov-Ma soliton dynamics in optical fibre , 2012, Scientific Reports.

[41]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[42]  Gino Biondini,et al.  Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions , 2006 .

[43]  N. Akhmediev,et al.  Modulation instability and periodic solutions of the nonlinear Schrödinger equation , 1986 .

[44]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[45]  Xin Zhou,et al.  Direct and inverse scattering transforms with arbitrary spectral singularities , 1989 .