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Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line

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Abstract

It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrödinger equation on the half-line can be expressed through the solution of a 2×2 matrix Riemann–Hilbert problem. This problem is specified by the spectral functions {a(k),b(k)} which are defined in terms of the initial condition q(x,0)=q 0(x), and by the spectral functions {A(k),B(k)} which are defined in terms of the specified boundary condition q(0,t)=g 0(t) and the unknown boundary value q x (0,t)=g 1(t). Furthermore, it has been shown that given q 0 and g 0, the function g 1 can be characterized through the solution of a certain 'global relation' coupling q 0, g 0, g 1, and Φ(t,k), where Φ satisfies the t-part ofthe associated Lax pair evaluated at x=0. We show here that, by using a Gelfand–Levitan–Marchenko triangular representation of Φ, the global relation can be explicitly solved for g 1.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, case 7012, Université Paris 7, 2 place Jussieu, 75251, Paris, France

    A. Boutet de Monvel & D. Shepelsky

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, U.K.

    A. S. Fokas

Authors
  1. A. Boutet de Monvel
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  2. A. S. Fokas
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  3. D. Shepelsky
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Boutet de Monvel, A., Fokas, A.S. & Shepelsky, D. Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line. Letters in Mathematical Physics 65, 199–212 (2003). https://doi.org/10.1023/B:MATH.0000010711.66380.77

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  • DOI: https://doi.org/10.1023/B:MATH.0000010711.66380.77

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