Persistence of periodic traveling waves and Abelian integrals

Armengol Gasull,Anna Geyer,Víctor Mañosa

doi:10.1016/j.jde.2021.05.033

Abstract

It is well known that the existence of traveling wave solutions (TWS) for many partial differential equations (PDE) is a consequence of the fact that an associated planar ordinary differential equation (ODE) has certain types of solutions defined for all time. In this paper we address the problem of persistence of TWS of a given PDE under small perturbations. Our main results deal with the situation where the associated ODE has a center and, as a consequence, the original PDE has a continuum of periodic traveling wave solutions. We prove that the TWS that persist are controlled by the zeroes of some Abelian integrals. We apply our results to several famous PDE, like the Ostrovsky, Klein-Gordon, sine-Gordon, Korteweg-de Vries, Rosenau-Hyman, Camassa-Holm, and Boussinesq equations.

Highlights

Traveling wave solutions (TWS) are an important class of particular solutions of partial differential equations (PDE). These waves are special solutions which do not change their shape and which propagate at constant speed. They appear in fluid dynamics, chemical kinetics involving reactions, mathematical biology, lattice vibrations in solid state physics, plasma physics and laser theory, optical fibers, etc
Recall that the boundary of the sets of structurally stable differential equations is precisely where bifurcations may occur. It is well-known that for many PDE the existence of TWS is established by proving the existence of a particular solution of a planar ordinary differential equation
It requires the existence of a certain wave speed c ∈ R such that: (i) the associated ordinary differential equations (ODE) has the form U ′′ = fc(U, U ′) + εgc(U, U ′, ε); (ii) after a time reparameterization if necessary the planar system associated with this ODE can be written as a perturbation of a Hamiltonian system; and (iii) this Hamiltonian system has a center, and the Melnikov-Poincare-Pontryagin function associated with the perturbation has l simple zeroes, see [4, Part II] for further details

Summary

Introduction

Traveling wave solutions (TWS) are an important class of particular solutions of partial differential equations (PDE). In a few words this means that if we fix a compact set K in the phase space it is said that an ODE is structurally stable on K when any other close enough (in the C1-topology) differential equation has a conjugated phase portrait This concept is relevant for applications because it implies that the observed behaviours are qualitatively robust with respect to small. Recall that the boundary of the sets of structurally stable differential equations is precisely where bifurcations (that is, qualitative changes of the phase portraits) may occur It is well-known that for many PDE the existence of TWS is established by proving the existence of a particular solution of a planar ordinary differential equation.

Definitions and main results

Some particular Abelian integrals

Applications