Existence of travelling-wave solutions to a coupled system of Korteweg–de Vries equations

The purpose of this paper is to provide a mathematical proof of the existence of smooth solitary travelling waves for a coupled system of Korteweg–de Vries type equations {ut+uxxx+(upvp+1)x=0,vt+rvx+vxxx+(up+1vp)x=0, where u=u(x,t),v=v(x,t)u=u(x,t),v=v(x,t) are real-valued functions and r,x,t∈Rr,x,t∈R. We follow a variational approach by characterizing travelling waves as minimizers of some functional under suitable constraints. Using the modern methods in the calculus of variations (the concentration-compactness principle of P. L. Lions), we prove that any minimizing sequence for the variational problem converges strongly, after an appropriate translation, to a minimizer.