Solutions with compact time spectrum to nonlinear Klein--Gordon and Schroedinger equations and the Titchmarsh theorem for partial convolution

TL;DR

This paper proves that finite energy solutions with compact time spectrum to certain nonlinear wave equations must be solitary waves, using a generalized Titchmarsh convolution theorem for partial convolutions.

Contribution

It introduces a novel application of the Titchmarsh theorem to partial convolutions, establishing a new criterion for solitary wave solutions.

Findings

Solutions with compact time spectrum are necessarily one-frequency solitary waves.

The approach generalizes the Titchmarsh convolution theorem to partial convolutions.

Provides a new analytical tool for studying nonlinear wave equations.

Abstract

We prove that finite energy solutions to the nonlinear Schroedinger 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.