The Three-Component Defocusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions

We present a rigorous theory of the inverse scattering transform (IST) for the three-component defocusing nonlinear Schrödinger (NLS) equation with initial conditions approaching constant values with the same amplitude as $${x\to\pm\infty}$$x→±∞. The theory combines and extends to a problem with non-zero boundary conditions three fundamental ideas: (i) the tensor approach used by Beals, Deift and Tomei for the n-th order scattering problem, (ii) the triangular decompositions of the scattering matrix used by Novikov, Manakov, Pitaevski and Zakharov for the N-wave interaction equations, and (iii) a generalization of the cross product via the Hodge star duality, which, to the best of our knowledge, is used in the context of the IST for the first time in this work. The combination of the first two ideas allows us to rigorously obtain a fundamental set of analytic eigenfunctions. The third idea allows us to establish the symmetries of the eigenfunctions and scattering data. The results are used to characterize the discrete spectrum and to obtain exact soliton solutions, which describe generalizations of the so-called dark-bright solitons of the two-component NLS equation.

[1]  S K Turitsyn,et al.  Vector dark solitons. , 1993, Optics letters.

[2]  Frédéric Dias,et al.  The Peregrine soliton in nonlinear fibre optics , 2010 .

[3]  M. Lakshmanan,et al.  Bright and dark soliton solutions to coupled nonlinear Schrodinger equations , 1995 .

[4]  Li-Chen Zhao,et al.  Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Gino Biondini,et al.  Elastic and inelastic line-soliton solutions of the Kadomtsev-Petviashvili II equation , 2005, Math. Comput. Simul..

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

[7]  A. Vartanian.  Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Non-Linear Schr\"{o}dinger Equation with Finite Density Initial Data. I. Solitonless Sector , 2001, nlin/0110024.

[8]  和達 三樹.  M. J. Ablowitz and H. Segur: Solitons and the Inverse Scattering Transform, Society for Industrial and Applied Mathematics, Philadelphia, 1981, x+425ページ, 23.5×16.5cm, $54.40 (SIAM Studies in Applied Mathematics). , 1982 .

[9]  C. Mee,et al.  The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions , 2014 .

[10]  M. Boiti,et al.  The spectral transform for the NLS equation with left-right asymmetric boundary conditions , 1982 .

[11]  Yuji Kodama,et al.  Young diagrams and N-soliton solutions of the KP equation , 2004, nlin/0406033.

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

[13]  Richard Squier,et al.  Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations. , 2003, Physical review letters.

[14]  Gino Biondini,et al.  The Integrable Nature of Modulational Instability , 2015, SIAM J. Appl. Math..

[15]  C. Hamner,et al.  Generation of dark-bright soliton trains in superfluid-superfluid counterflow. , 2010, Physical review letters.

[16]  P. Drazin.  SOLITONS, NONLINEAR EVOLUTION EQUATIONS AND INVERSE SCATTERING (London Mathematical Society Lecture Note Series 149) , 1993 .

[17]  Yuji Kodama,et al.  The Deodhar decomposition of the Grassmannian and the regularity of KP solitons , 2012, 1204.6446.

[18]  D. J. Kaup,et al.  The Three-Wave Interaction-A Nondispersive Phenomenon , 1976 .

[19]  P. Deift,et al.  Inverse scattering on the line , 1979 .

[20]  A. Vartanian,et al.  Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua , 2002 .

[21]  C. Hamner,et al.  Dark-dark solitons and modulational instability in miscible two-component Bose-Einstein condensates , 2010, 1007.4947.

[22]  T. Frankel.  The geometry of physics : an introduction , 2004 .

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

[24]  Yuri S. Kivshar,et al.  Polarized dark solitons in isotropic Kerr media , 1997 .

[25]  Y. Kodama,et al.  KP solitons, total positivity, and cluster algebras , 2011, Proceedings of the National Academy of Sciences.

[26]  David W. Lewis,et al.  Matrix theory , 1991 .

[27]  Leon A. Takhtajan,et al.  Hamiltonian methods in the theory of solitons , 1987 .

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

[29]  Gino Biondini,et al.  Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions , 2014 .

[30]  William E. Schiesser,et al.  Linear and nonlinear waves , 2009, Scholarpedia.

[31]  B. Prinari,et al.  Dark–dark and dark–bright soliton interactions in the two-component defocusing nonlinear Schrödinger equation , 2013 .

[32]  K. Steiglitz,et al.  Energy-Exchange Interactions between Colliding Vector Solitons , 1999 .

[33]  G. Kovačič,et al.  The focusing Manakov system with nonzero boundary conditions , 2015 .

[34]  P. G. Kevrekidis,et al.  Beating dark–dark solitons in Bose–Einstein condensates , 2012, 1202.2777.

[35]  Gino Biondini,et al.  On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy , 2003, nlin/0306003.

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

[37]  Helge Holden,et al.  Soliton Equations and Their Algebro-Geometric Solutions: The AKNS Hierarchy , 2003 .

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

[39]  Y. Kodama,et al.  Submitted to: J. Phys. A: Math. Gen. , 2003 .

[40]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[41]  Gino Biondini,et al.  Inverse Scattering Transform for the Defocusing Manakov System with Nonzero Boundary Conditions , 2015, SIAM J. Math. Anal..

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

[43]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[44]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .

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

[46]  R. Sachs.  Review: Richard Beals, Percy Deift and Carlos Tomei, Direct and inverse scattering on the line , 1990 .

[47]  M. Ablowitz,et al.  Soliton interactions in the vector NLS equation , 2004 .

[48]  D. Mihalache,et al.  Multichannel soliton transmission and pulse shepherding in bit-parallel-wavelength optical fiber links , 2002 .

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

[50]  B. Prinari,et al.  The Three-Component Defocusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions , 2015, Communications in Mathematical Physics.

[51]  Nonlinear Waves, Solitons and Chaos. By E. I NFELD and G. R OWLANDS . Cambridge University Press, 1990. 423 pp. £45 or $85 (hardback), £17.50 or $29.95 (paperback). , 1991 .

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

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

[54]  A. Ustinov.  Asymptotic solution of the Cauchy problem for the nonlinear Schrödinger equation with boundary conditions of finite density type , 1989 .

[55]  A. Vartanian,et al.  Exponentially small asymptotics of solutions to the defocusing nonlinear Schrödinger equation: II , 2001 .

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

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

[58]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

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

[60]  Q. Park,et al.  Systematic construction of multicomponent optical solitons , 2000 .

[61]  J. R. Ensher,et al.  Dynamics of component separation in a binary mixture of Bose-Einstein condensates , 1998 .

[62]  Gino Biondini,et al.  Soliton solutions of the Kadomtsev-Petviashvili II equation , 2006 .

[63]  K. Nakkeeran,et al.  Exact dark soliton solutions for a family of N coupled nonlinear Schrödinger equations in optical fiber media. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[64]  G. B. Whitham,et al.  Linear and Nonlinear Waves: Whitham/Linear , 1999 .

[65]  Gino Biondini,et al.  Line soliton interactions of the Kadomtsev-Petviashvili equation. , 2007, Physical review letters.

[66]  G. Rowlands,et al.  Nonlinear Waves, Solitons and Chaos , 1990 .

[67]  E. Belokolos,et al.  Algebro-geometric approach to nonlinear integrable equations , 1994 .

[68]  A.H. Vartanian,et al.  Exponentially small asymptotics of solutions to the defocusing nonlinear Schrödinger equation , 2003, Appl. Math. Lett..

[69]  B. Prinari,et al.  Inverse scattering transform for the defocusing nonlinear Schrödinger equation with fully asymmetric non-zero boundary conditions , 2016 .

[70]  S. V. Manakov.  On the theory of two-dimensional stationary self-focusing of electromagnetic waves , 1973 .

[71]  Kenneth Steiglitz,et al.  State transformations of colliding optical solitons and possible application to computation in bulk media , 1998 .

[72]  Helge Holden,et al.  Soliton Equations and Their Algebro-Geometric Solutions: Bibliography , 2003 .

[73]  Carlos Tomei,et al.  Direct and inverse scattering on the line , 1988 .

[74]  Yasuhiro Ohta,et al.  General N‐Dark–Dark Solitons in the Coupled Nonlinear Schrödinger Equations , 2011 .

[75]  Yuji Kodama,et al.  Classification of the line-soliton solutions of KPII , 2007, 0710.1456.

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

[77]  Pelinovsky,et al.  Stability criterion for multicomponent solitary waves , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[79]  B. Prinari,et al.  Polarization interactions in multi-component defocusing media , 2015 .

[80]  Fabio Baronio,et al.  Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. , 2012, Physical review letters.

[81]  K. Bongs,et al.  Oscillations and interactions of dark and dark bright solitons in Bose Einstein condensates , 2008 .