Existence of travelling-wave solutions to a coupled system of Korteweg–de Vries equations. (November 2015) - Bhattarai, Santosh
Abstract: 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 + u x x x + ( u p v p + 1 ) x = 0, v t + r v x + v x x x + ( u p + 1 v p ) x = 0, where u = u ( x, t ), v = v ( x, t ) are real-valued functions and r, 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.