2003/07/25 by P. G. Grinevich, P. M. Santini
Physics and Astronomy · Mathematics · #nlin.SI #hep-th #math-ph #math.AG #math.AP #math.MP
published as American Mathematical Society Translations - Series 2, Advances in the Mathematical Sciences, 2004, v. 212., pp. 157-178. · 29 pages, LaTeX, 2 Encapsulated Postscript figures
arxiv created 2003/07/25 · arxiv updated 2009/12/01
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.