Solvability of a mixed problem for the nonlinear Schrödinger equation. (English. Russian original) Zbl 0658.35017

Vladimirov, M. V.

Math. USSR, Sb. 58, 525-540 (1987); translation from Mat. Sb., Nov. Ser. 130(172), No. 4, 520-536 (1986).

The existence and uniqueness of the solution of a mixed problem for the nonlinear Schrödinger equation of the form \[ \partial u/\partial t+i\Delta u+i\alpha | u|^ pu+\beta | u|^ qu=0,\quad (x,t)\in Q=G\times [0,T], \] at the boundary condition \(u|_{t=0}=u_ 0(x)\) is studied. In this equation G is a finite region of an n-dimensional space \(R^ n\), \(\alpha\) is a real number in the spaces \(L_{\infty}(0,T;\overset\circ W^ 1_ 2(G)\) and \(L_{\infty}(0,T;\overset\circ W^ 1_ 2(G)\cap W^ 2_ 2(G)\) at \(q\geq p\geq 0\), \(\beta >0\). The essential results were already published by the author [Sov. Math. Dokl. 29, 281-284 (1984); translation from Dokl. Akad. Nauk SSSR 275, 780-783 (1984; Zbl 0585.35019)]. The existence of the generalized solution was proved by the Galerkin method under additional conditions imposed on n, p, q, \(\alpha\) and \(\beta\). The asymptotics of the solution are given. The uniqueness of the solution is also proved. Such an equation describes e.g. light beam propagation in nonlinear media

Reviewer: V.Burjan

MSC:

35G30	Boundary value problems for nonlinear higher-order PDEs
35J10	Schrödinger operator, Schrödinger equation
78A10	Physical optics
35A35	Theoretical approximation in context of PDEs
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

existence; uniqueness; nonlinear Schrödinger equation; generalized solution; Galerkin method; asymptotics; light beam propagation; nonlinear media

Citations:

Zbl 0585.35019