The initial boundary value problem on the segment for the Nonlinear Schr\"odinger equation; the algebro-geometric approach. I
Grinevich, P. G. ; Santini, P. M.
arXiv, 0307026 / Harvested from arXiv
Text on arXiv
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 Schr\"odinger 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.
Publié le : 2003-07-16
Classification:  Nonlinear Sciences - Exactly Solvable and Integrable Systems,  High Energy Physics - Theory,  Mathematical Physics,  Mathematics - Algebraic Geometry,  Mathematics - Analysis of PDEs 
@article{0307026,
     author = {Grinevich, P. G. and Santini, P. M.},
     title = {The initial boundary value problem on the segment for the Nonlinear
  Schr\"odinger equation; the algebro-geometric approach. I},
     journal = {arXiv},
     volume = {2003},
     number = {0},
     year = {2003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0307026}
}

Grinevich, P. G.; Santini, P. M. The initial boundary value problem on the segment for the Nonlinear
  Schr\"odinger equation; the algebro-geometric approach. I. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0307026/