Dynamics of Partial Differential Equations

Volume 6 (2009)

Number 4

Hylomorphic solitons in the nonlinear Klein-Gordon equation

Pages: 311 – 334

DOI: https://dx.doi.org/10.4310/DPDE.2009.v6.n4.a2

Authors

J. Bellazzini (Dipartimento di Matematica Applicata, Università degli Studi di Pisa, Italy)

V. Benci (Dipartimento di Matematica Applicata, Università degli Studi di Pisa, Italy)

C. Bonanno (Dipartimento di Matematica Applicata, Università degli Studi di Pisa, Italy)

E. Sinibaldi (Italian Institute of Technology (IIT), Center for Micro-Biorobotics, Pontedera, Pisa, Italy)

Abstract

Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localised packet and which preserves this localisation in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behaviour. In this paper we show a new mechanism which might produce solitary waves and solitons for a large class of equations, such as the nonlinear Klein-Gordon equation. We show that the existence of these kind of solitons, that we have called hylomorphic solitons, depends on a suitable energy/charge ratio. We show a variational method that allows to prove the existence of hylomorphic solitons and that turns out to be very useful for numerical applications. Moreover we introduce some classes of nonlinearities which admit hylomorphic solitons of different shapes and with different relations between charge, energy and frequency.

Keywords

hylomorphic solitons, nonlinear Klein-Gordon equation

2010 Mathematics Subject Classification

35-xx, 76-xx

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Published 1 January 2009