Solitary waves of the generalized KdV equation with distributed delays

Abstract

In this paper, we study an integro-differential equation based on the generalized KdV equation with a convolution term which introduces a time delay in the nonlinearity. Special attention is paid to the existence of solitary wave solutions. Motivated by [M.J. Ablowitz, H. Seger, Soliton and Inverse Scattering Transform, SIAM, Philadelphia, 1981; C.K.R.T. Jones, Geometrical singular perturbation theory, in: R. Johnson (Ed.), Dynamical Systems, in: Lecture Notes in Math., vol. 1609, Springer, New York, 1995; T. Ogawa, Travelling wave solutions to perturbed Korteweg–de Vries equations, Hiroshima Math. J. 24 (1994) 401–422], we prove, using the linear chain trick and geometric singular perturbation analysis, that the solitary wave solutions persist when the average delay is suitably small, for a special convolution kernel.