The direct and inverse scattering problem for the semilinear Schrödinger equation

Abstract

We study the direct and inverse scattering problem for the semilinear Schrödinger equation $\Delta u+a(x,u)+k^{2}u=0$ in ${\mathbb{R}}^d$. We show well-posedness in the direct problem for small solutions based on the Banach fixed point theorem, and the solution has the certain asymptotic behavior at infinity. We also show the inverse problem that the semilinear function $a(x,z)$ is uniquely determined from the scattering data. The idea is certain linearization that by using sources with several parameters we differentiate the nonlinear equation with respect to these parameter in order to get the linear one.

19 pages

References in corpus (1)

• Inverse scattering on the line for a generalized nonlinear Schroedinger equation

Cited by in corpus (1)

• Inverse Scattering for Critical Semilinear Wave Equations