On the compactness of the support of solitary waves of the complex saturated nonlinear Schr{\"o}dinger equation and related problems

TL;DR

This paper proves the compactness of the support of solitary wave solutions to a complex saturated nonlinear Schr{"o}dinger equation, including existence, uniqueness, and the behavior under various boundary conditions and couplings.

Contribution

It establishes the first rigorous proof of solitons with compact support for this class of nonlinear Schr{"o}dinger equations, extending previous claims without proof.

Findings

Support of solutions is compact under certain conditions.

Existence and uniqueness of solutions are confirmed.

Solutions can have compact support even with non-vanishing sources.

Abstract

We study the vectorial stationary Schr{\"o}dinger equation -$\Delta$u + a U + b u = F, with a saturated nonlinearity U = u/|u| and with some complex coefficients (a, b) $\in$ C 2 . Besides the existence and uniqueness of solutions for the Dirichlet and Neumann problems, we prove the compactness of the support of the solution, under suitable conditions on (a, b) and even when the source in the right hand side F (x) is not vanishing for large values of |x|. The proof of the compactness of the support uses a local energy method, given the impossibility of applying the maximum principle. We also consider the associate Schr{\"o}dinger-Poisson system when coupling with a simple magnetic field. Among other consequences, our results give a rigorous proof of the existence of ''solitons with compact support'' claimed, without any proof, by several previous authors.