The initial boundary value problem on the segment for the Nonlinear Schrödinger equation; the algebro-geometric approach. I

P. G. Grinevich; P. M. Santini

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The initial boundary value problem on the segment for the Nonlinear Schrödinger equation; the algebro-geometric approach. I

P. G. Grinevich·P. M. Santini

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Abstract

This is the first of a series of papers devoted to the study of classical initial-boundary value problems of Dirichlet, Neumann and mixed type for the Nonlinear Schrödinger equation on the segment. Considering proper periodic discontinuous extensions of the profile, generated by suitable point-like sources, we show that the above boundary value problems can be rewritten as nonlinear dynamical systems for suitable sets of algebro-geometric spectral data, generalizing the classical Dubrovin equations. In this paper we consider, as a first illustration of the above method, the case of the Dirichlet problem on the segment with zero-boundary value at one end, and we show that the corresponding dynamical system for the spectral data can be written as a system of ODEs with algebraic right-hand side.

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P. G. Grinevich, P. M. Santini. 2003-07-25. The initial boundary value problem on the segment for the Nonlinear Schrödinger equation; the algebro-geometric approach. I. https://arxiv.org/abs/nlin/0307026

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