A minimization method and applications to the study of solitons

TL;DR

This paper introduces a general minimization framework for establishing the existence of stable, particle-like solitons called hylomorphic solitons, and applies it to nonlinear Schrödinger and Klein-Gordon equations.

Contribution

It develops an abstract theorem for proving soliton existence via constrained minimization and demonstrates its application to key nonlinear wave equations.

Findings

Established a general theorem for soliton existence

Applied the theory to nonlinear Schrödinger and Klein-Gordon equations

Proved the existence of hylomorphic solitons in these contexts

Abstract

Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behavior. In this paper, we prove a general, abstract theorem (Theorem 26) which allows to prove the ex istence of a class of solitons. Such solitons are suitable minimizers of a constrained functional and they are called hylomorphic solitons. Then we apply the abstract theory to problems related to the nonlinear Schr\"odinger equation (NSE) and to the nonlinear Klein-Gordon equation (NKG).