Persistence of periodic traveling waves and Abelian

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:293

Article

Gasull, Armengol^1,2  Geyer, Anna³  Manosa, Victor^4,5 

[1] Univ Autonoma Barcelona, Dept Matemat, Edif Cc, Barcelona 08193, Spain

[2] Ctr Recerca Matemat, Edif Cc,Campus Bellaterra, Barcelona 08193, Spain

[3] Delft Univ Technol, Delft Inst Appl Math, Fac Elect Engn Math & Comp Sci, Van Mourik Broekmanweg 6, NL-2628 XE Delft, Netherlands

[4] Univ Politecn Cataluna, Dept Matemat, Colom 11, Terrassa 08222, Spain

[5] Univ Politecn Cataluna, Inst Matemat, UPC BarcelonaTech IMTech, Colom 11, Terrassa 08222, Spain

关键词: Traveling wave;    Abelian integral;    Melnikov-Poincare-Pontryagin function;    Periodic orbit;    Limit cycle;    Bifurcation;   

DOI  :  10.1016/j.jde.2021.05.033

来源: Elsevier

【 摘 要 】

It is well known that the existence of traveling wave solutions (TWS) for many partial differential equa-tions (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 asso-ciated 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. (c) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

【 授权许可】

Free