Abstract
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 \({x\to\pm\infty}\). 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, vol. 149. Cambridge University Press, Cambridge (1992)
Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, vol. 302. Cambridge University Press, Cambridge (2004)
Ablowitz M.J., Segur H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)
Baronio F., Degasperis A., Conforti M., Wabnitz S.: Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. Phys. Rev. Lett. 109, 044012 (2012)
Beals R., Coifman R.R.: Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math. 37, 39–90 (1984)
Beals R., Deift P., Tomei C.: Direct and Inverse Scattering on the Line. American Mathematical Society, Providence (1988)
Belokolos E.D., Bobenko A.I., Enol’skii V.Z., Its A.R., Matveev V.B.: Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer, Berlin (1994)
Biondini G.: Soliton interactions in the Kadomtsev–Petviashvili II equation. Phys. Rev. Lett. 99, 064103 (2007)
Biondini G., Chakravarty S.: Soliton solutions of the Kadomtsev–Petviashvili II equation. J. Math. Phys. 47, 033514 (2006)
Biondini G., Chakravarty S.: Elastic and inelastic line-soliton solutions of the Kadomtsev–Petviashvili II equation. Math. Comput. Simul. 74, 237 (2007)
Biondini G., Fagerstrom E.R.: The integrable nature of modulational instability. SIAM J. Appl. Math. 75, 136–163 (2015)
Biondini, G., Fagerstrom, E.R., Prinari, B.: Inverse scattering transform for the defocusing nonlinear Schrödinger equation with asymmetric boundary conditions. Phys. D (2016, to appear)
Biondini G., Kodama Y.: On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hiera rchy. J. Phys. A 36, 10519–10536 (2003)
Biondini G., Kovacic G.: Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions. J. Math. Phys. 55, 031506 (2014)
Biondini G., Kraus D.K.: Inverse scattering transform for the defocusing Manakov system with nonzero boundary conditions. SIAM J. Math. Anal. 47, 706–757 (2015)
Biondini G., Kraus D.K., Prinari B., Vitale F.: Polarization interactions in multi-component repulsive Bose–Einstein condensates. J. Phys. A. 48, 395202 (2015)
Biondini G., Prinari B.: On the spectrum of the Dirac operator and the existence of discrete eigenvalues for the defocusing nonlinear Schrödinger equation. Stud. Appl. Math. 132, 138–159 (2014)
Boiti M., Pempinelli F.: The spectral transform for the NLS equation with left–right asymmetric boundary conditions. Nuovo Cimento A 69, 213–227 (1982)
Chakravarty S., Kodama Y.: Classification of the soliton solutions of KPII. J. Phys. A 41, 275209 (2008)
Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Spproach, Courant Lecture Notes, vol. 3. Courant Institute of Mathematical Sciences (2000)
Deift P., Trubowitz E.: Inverse scattering on the line. Commun. Pure Appl. Math. 21, 121–251 (1979)
Deift P., Venakides S., Zhou X.: The collisionless shock region for the long-time behavior of solutions of the KdV equation. Commun. Pure Appl. Math. 47, 199–206 (1994)
Deift P., Zhou X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the mKdV equation. Ann. Math. 137, 295–368 (1993)
Demontis F., Prinari B., van der Mee C., Vitale F.: The inverse scattering transform for the defocusing nonlinear Schrödinger equations with nonzero boundary conditions. Stud. Appl. Math. 131, 1–40 (2013)
Demontis F., Prinari B., van der Mee C., Vitale F.: The inverse scattering transform for the focusing nonlinear Schrodinger equation with asymmetric boundary conditions. J. Math. Phys. 55, 101505 (2014)
Faddeev L.D., Takhtajan L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)
Frankel T.: The Geometry of Physics: An Introduction. Cambridge University Press, Cambridge (2012)
Gantmacher F.R.: Matrix Theory, vol. 1. American Mathematical Society, Providence (2000)
Gesztesy F., Holden H.: Soliton Equations and Their Algebro-geometric Solutions. Cambridge University Press, Cambridge (1990)
Hoefer M.A., Chang J.J., Hamner C., Engels P.: Dark-dark solitons and modulational instability in miscible two-component Bose–Einstein condensates. Phys. Rev. A 84, 041605 (2011)
Infeld E., Rowlands G.: Nonlinear Waves, Solitons and Chaos. Cambridge University Press, Cambridge (2003)
Its A.R., Ustinov A.F.: Temporal asymptotic solution of the Cauchy problem for the nonlinear Schrödinger equation with boundary conditions of the finite-density type. Sov. Phys. Dokl. 31, 893–895 (1986)
Kaup D.J.: The three-wave interaction—a nondispersive phenomenon. Stud. Appl. Math. 55, 9–44 (1976)
Kibler B., Fatome J., Finot C., Millot G., Dias F., Genty G., Akhmediev N., Dudley J.M.: The Peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790–795 (2010)
Kodama Y.: Young diagrams and N-soliton solutions of the KP equation. J. Phys. A 37, 11169 (2004)
Kodama Y., Williams L.: KP solitons, total positivity, and cluster algebras. Proc. Nat. Acad. Sci. 108, 8984–8989 (2011)
Kodama Y., Williams L.: The Deodhar decomposition of the Grassmannian and the regularity of KP solitons. Adv. Math. 244, 979–1032 (2013)
Kraus D.K., Biondini G., Kovacic G.: The focusing Manakov system with nonzero boundary conditions. Nonlinearity 28, 3101–3151 (2015)
Lax P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467 (1968)
Manakov S.V.: On the theory of two-dimensional stationary self-focusing electromagnetic waves. Sov. Phys. JETP 38, 248–253 (1974)
Novikov S.P., Manakov S.V., Pitaevskii L.P., Zakharov V.E.: Theory of Solitons: The Inverse Scattering Method. Plenum, New York (1984)
Prinari B., Ablowitz M.J., Biondini G.: Inverse scattering transform for the vector nonlinear Schrödinger equation with non-vanishing boundary conditions. J. Math. Phys. 47, 063508 (2006)
Prinari B., Biondini G., Trubatch A.D.: Inverse scattering transform for the multi-component nonlinear Schrödinger equation with nonzero boundary conditions. Stud. Appl. Math. 126, 245–302 (2011)
Prinari B., Vitale F., Biondini G.: Dark-bright soliton solutions with nontrivial polarization interactions for the three-component defocusing nonlinear Schrödinger equation with non-zero boundary conditions. J. Math. Phys. 56, 071505 (2015)
Sulem C., Sulem P.L.: The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse. Springer, New York (1999)
Vartanian A.H.: Exponentially small asymptotics of solutions to the defocusing nonlinear Schrödinger equation. J. Phys. A Math Gen. 34, L647–L655 (2001)
Vartanian A.H.: Long-time asymptotics of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation with finite-density initial data. Math. Phys. Anal. Geom. 5, 319–413 (2002)
Whitham G.B.: Linear and Nonlinear Waves. Wiley, New York (1999)
Yan D., Chang J.J., Hamner C., Hoefer M., Kevrekidis P.G., Engels P., Achilleos V., Frantzeskakis D.J., Cuevas J.: Beating dark-dark solitons in Bose–Einstein condensates. J. Phys. B 45, 115301 (2012)
Zakharov V.E., Ostrovsky L.A.: Modulation instability: the beginning. Phys. D 238, 540–548 (2009)
Zakharov V.E., Shabat A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972)
Zakharov V.E., Shabat A.B.: Interaction between solitons in a stable medium. Sov. Phys. JETP 37, 823–828 (1973)
Zhao L.C., Liu J.: Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation. Phys. Rev. E 87, 013201 (2013)
Zhou X.: Direct and inverse scattering transforms with arbitrary spectral singularities. Commun. Pure Appl. Math. 42, 895–938 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Deift
Rights and permissions
About this article
Cite this article
Biondini, G., Kraus, D.K. & Prinari, B. The Three-Component Defocusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions. Commun. Math. Phys. 348, 475–533 (2016). https://doi.org/10.1007/s00220-016-2626-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-016-2626-7