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The initial-boundary value problem on the interval for the nonlinear Schrödinger equation. The algebro-geometric approach. I. (English) Zbl 1076.37061

Buchstaber, V. M. (ed.) et al., Geometry, topology, and mathematical physics. Selected papers from S. P. Novikov’s seminar held in Moscow, Russia, 2002–2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3613-7/hbk). Translations. Series 2. American Mathematical Society. 212. Advances in the Mathematical Sciences 55, 157-178 (2004).
This is the first of the authors’s papers contributed to solving classical initial-boundary value problems of Dirichlet, Neumann and mixed type for the nonlinear Schrödinger (NLS) equation on the finite interval. The authors aim to show that the Dirichlet, Neumann and mixed type problems of the NLS equation on the finite interval can be transformed into periodic Cauchy problems on the line with point-like \(\delta \) and/or \(\delta '\) sources, and thus, using the standard algebro-geometric approach, these problems can be reduced to nonlinear systems of ordinary differential equations describing the time evolutions of the spectral data of the transformed periodic problems with forcings. This paper particularly studies a Dirichlet problem of the defocussing NLS equation on the finite interval with zero boundary value at one end. The spectral characterization of the discontinuous periodic profiles arising from the Dirichlet problem is presented, and the corresponding nonlinear system of ordinary differential equations are derived. A few concluding remarks on the presented approach are given at the end of the paper.
For the entire collection see [Zbl 1051.00009].
Reviewer: Ma Wen-Xiu (Tampa)

MSC:

37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q55 NLS equations (nonlinear Schrödinger equations)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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