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 (u(t) therefore broken vertical bar u(xxx) broken vertical bar (u(p)upsilon(p+1))(x) = 0, (upsilon(t) broken vertical bar rv(x) broken vertical bar u(xxx) broken vertical bar (u(p)upsilon(p+1))(x) = 0, where u = u(x,t), nu = nu(x, t) are real-valued functions and r, x, t E 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. (C) 2015 Elsevier Ltd. All rights reserved.