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

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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.

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Authors and Affiliations

  1. Department of Mathematics, State University of New York at Buffalo, Buffalo, NY, USA

    Gino Biondini & Daniel K. Kraus

  2. Department of Mathematics, University of Colorado Colorado Springs, Colorado Springs, CO, USA

    Barbara Prinari

  3. Department of Mathematics and Physics, and Sezione INFN, Università del Salento, Lecce, Italy

    Barbara Prinari

Authors
  1. Gino Biondini
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  2. Daniel K. Kraus
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  3. Barbara Prinari
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Correspondence to Gino Biondini.

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Communicated by P. Deift

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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

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