Solitary wave solutions for a generalized KdV–mKdV equation with distributed delays∗

In the past three decades, traveling wave solutions to the Korteweg–de Vries equation have been studied extensively and a large number of theoretical issues concerning the KdV equation have received considerable attention. These wave solutions when they exist can enable us to well understand the mechanism of the complicated physical phenomena and dynamical processes modeled by these nonlinear evolution equations. One can easily find abundant reports about it, such as [1–7]. And many powerful methods to construct exact solutions of KdV have been established and developed. Among these methods we mainly cite, for example, the bifurcation method of dynamical systems [8], (G′/G)-expansion method [9–11] and the sub-ODE method [12], Lie group theoretical methods [13] and so on. In this paper, we will use the geometrical singular perturbation theory and the linear chain trick to investigate solitary wave solutions of the generalized KdV–mKdV equation. Let us consider a generalized Korteweg–de Vries–modified Korteweg–de Vries (KdV– mKdV) equation Ut + ( α+ βU + γU ) Ux + Uxxx = 0, (1)