Existence of travelling-wave solutions to a coupled system of Korteweg–de Vries equations. (November 2015)

• Record Type: Journal Article
• Authors: Bhattarai, Santosh
• Is Part Of: Nonlinear analysis. Volume 127(2015)
• Page Start: 182
• Page End: 195
• Publication Date: 2015-11
• Subjects: primary 35Q53 -- secondary 35A15 35C07; Travelling-wave solutions -- Existence -- Coupled KdV equations -- Concentration-compactness method; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.07.004
• Languages: English
• ISSNs: 0362-546X

Abstract

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.