Travelling Waves For Two Coupled Korteweg-de Vries Equations
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F. P. Barrera & T. Brugarino
The aim of this paper is to apply the auxiliary equation method to obtain solutions of the system of two coupled Korteweg-de Vries equations which have many applications in the study of two-wave modes in stratified liquid. As an auxiliary equation we consider a reducible hyperelliptic equation. Keywords: non linear equations, exact solutions, KdV equations. 1 Introduction We consider the two coupled Korteweg-de Vries equations: ut + α1ux + α2vx + α3uux + α4uxxx = 0 vt + β1vx + β2ux + β3vvx + β4vxxx = 0 (1) which have many applications in the study of two-wave modes in stratified liquid [1–3]. Recently a large number of methods, such as the tanh method [4,5], generalized tanh method [6,7] and the Jacobi elliptic function expansionmethod , have been proposed to construct explicit analytical solutions for nonlinear wave equations. Using the reducible hyperelliptic equation : p2 ξ (ξ ) = a0 + a2p2(ξ ) + a4p4(ξ ) + a6p6(ξ ) (2) we obtain travelling wave solutions of Eqns (1). These solutions are expressed in terms of the Weirestrass elliptic function ℘.
non linear equations, exact solutions, KdV equations.