Formulation of the scattering theory for the NLS equation with boundary conditions of the type of finite density in the soliton-free sector. (Russian. English summary) Zbl 0695.35145

As in an earlier paper [see the authors, Sov. Phys., Dokl. 31, 893-895 (1986); translation from Dokl. Akad. Nauk SSSR 291, 91-95 (1986; Zbl 0651.35015)] the authors consider the Cauchy problem for the one- dimensional nonlinear Schrödinger equation with initial values, which satisfy so-called “finite density boundary conditions” at infinity. The aim of the paper is to calculate the time-asymptotic behavior of the associated symplectic form. Under the assumption that there is no discrete spectrum (in the case of a “soliton-free sector”) they find the proper asymptotic variables and thus they explain the occurrence of some phenomena known in the scattering theory of the Korteweg-de Vries equation, see e.g., V. S. Buslaev, L. D. Faddeev and L. A. Takhtadzhan [Physica D 18, 255-266 (1986; Zbl 0618.35100)].
Reviewer: E.Lanckau


35P25 Scattering theory for PDEs
81U20 \(S\)-matrix theory, etc. in quantum theory
35Q99 Partial differential equations of mathematical physics and other areas of application
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