On solutions with compact spectrum to nonlinear Klein--Gordon and Schroedinger equations

TL;DR

This paper investigates conditions under which solitary waves with compact spectrum are unique solutions in nonlinear Klein-Gordon and Schrödinger equations, and constructs multi-frequency solutions in nonlinear Dirac equations.

Contribution

It identifies conditions ensuring the uniqueness of one-frequency solitary waves and provides an example of multi-frequency solutions in nonlinear Dirac equations.

Findings

Standard solitary waves are unique under certain nonlinearities.

Constructed a four-frequency solitary wave in nonlinear Dirac equations.

Clarified spectral properties of solutions in nonlinear wave equations.

Abstract

We consider finite energy solutions to the nonlinear Schroedinger equation and nonlinear Klein--Gordon equation and find the condition on the nonlinearity so that the standard, one-frequency solitary waves are the only solutions with compact spectrum. We also construct an example of a four-frequency solitary wave solution to the nonlinear Dirac equation in three dimensions.