Solutions with Compact Time Spectrum to Nonlinear Klein–Gordon and Schrödinger Equations and the Titchmarsh Theorem for Partial Convolution

Abstract

We prove that finite energy solutions to the nonlinear Schrödinger equation and nonlinear Klein-Gordon equation which have the compact time spectrum have to be one-frequency solitary waves. The argument is based on the generalization of the Titchmarsh convolution theorem to partial convolutions.

A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II

“…Global asymptotic results have been proved for equations where the nonlinearity is concentrated in a point, or in finitely many points, that is β(|u| 2 )u is replaced by δ(x − x 0 )β(|u| 2 )u, or by a linear combination of such terms. See [82]- [85], [22] and therein.…”