Existence of Generalized Solitary Wave Solutions of the Coupled KdV-CKdV System

This paper studies the traveling wave solutions of the coupled KdV-CKdV system {u(t) + 2bu(xi) + au(xi xi xi) = -2b(uv)(xi), v(t) + bv(xi) + bvv(xi) + cv(xi xi xi) = -b(vertical bar u vertical bar(2))(xi), where the parameters a, b, c are real. If these parameters satisfy some conditions, the origin is a saddle-center equilibrium, that is, the linear operator at the origin has a pair of positive and negative eigenvalues and a pair of purely imaginary eigenvalues where the real eigenvalues bifurcate from a double eigenvalue 0. We first change this system with a travelingwave frame into an ordinary differential systemwith dimension 4, and then give the homoclinic solution of its dominant system and the periodic solution of the whole system if the first mode in the Fourier series of the function v is activated, respectively. Using the fixed point theorem, the perturbation methods, and the reversibility, we rigorously prove that this homoclinic solution, when higher order terms are added, will persist and exponentially tend to the obtained periodic solution (called generalized homoclinic solution), which presents the existence of the generalized solitary wave solution (solitary wave solution exponentially approaching a periodic solution).