Resumen de Existence of Single-Peak Solitary Waves and Double-Peaks Solitary Wave of Gardner Equation with Kuramoto–Sivashinsky Perturbation

Keqin Zhao, Zhenshu Wen

• This paper concerns geometric study of single-peak solitary waves and double-peaks solitary wave of Gardner equation with Kuramoto–Sivashinsky perturbation. We first reduce the high-dimensional traveling wave system of the perturbed Gardner equation to the perturbed planar system through geometric singular perturbation theory. We then show the persistence of one homoclinic orbit, and the generation of a new homoclinic orbit by the Melnikov function method. Single-peak solitary waves and double-peaks solitary wave are newly found for the perturbed Gardner equation. The numerical simulations are performed to verify the theoretical results.