Persistence of solitary wave solutions of singularly perturbed Gardner equation Yaning Tang a , Wei Xu a, * , Jianwei Shen a,b , Liang Gao a a Department of Applied Mathematics, Northwestern Polytechnical University Xi’an, Shaanxi 710072, PR China b Department of Mathematics, Xuchang University, Xuchang, Henan 461000, PR China Accepted 5 September 2006 Communicated by Prof. Ji-Huan He Abstract The paper studies the singularly perturbed Gardner equation. Based on the relation between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, the persistence of the solitary wave solution for the singularly perturbed Gardner equation is investigated using a geometric singular perturbation method. We show that the solitary wave solution exists when the perturbation parameter is sufficiently small. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction There are many methods for obtaining explicit and exact traveling wave solutions [1–7] of nonlinear wave equations. However, when talking about perturbed wave equations, especially singularly perturbed ones, the first question is the existence of traveling wave solutions. This problem has been investigated by several authors [8–14], here the geometric singular perturbation methods [10,11] plays a special role in giving a first picture of the perturbed solutions. In this paper, we consider existence of the solitary wave of the singularly perturbed Gardner equation / t þ c// x þ a/ 2 / x þ b/ xxx þ eð/ xx þ / xxxx Þ¼ 0; ð1Þ where c, a, b, e are positive parameters. 2. Geometric singular perturbation theory In this section, we first introduce the geometric singular perturbation theory [10,11]. Consider the standard fast-slow system 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.044 * Corresponding author. Tel./fax: +86 29 88495453. E-mail addresses: tyaning@nwpu.edu.cn (Y. Tang), weixu@nwpu.edu.cn (W. Xu). Available online at www.sciencedirect.com Chaos, Solitons and Fractals 37 (2008) 532–538 www.elsevier.com/locate/chaos