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

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)

Zbl 0585.35019
Full Text: