On Solitary-Wave Solutions for the Coupled Korteweg – de Vries and Modified Korteweg – de Vries Equations and their Dynamics

which can be considered as a coupling between the KdV (with respect to u) and the mKdV (with respect to v) equations. The coupled KdV-mKdV equations were proposed by Kersten and Krasil’shchik [1] and originate from a supersymmetric extension of the classical KdV [2]. It also can be considered as a coupling between the KdV and mKdV equations: By setting v = 0 we obtain the KdV equation ut + uxxx − 6uux = 0; by setting u = 0, we obtain the mKdV equation vt + vxxx − 3vvx = 0. In here, (uv)x acts as a force term on the first KdV equation, which is coupled to the second equation of similar type, without any dispersion term. The complete integrability of the coupled KdVmKdV equations was shown by Kersten and Krasil’shchik by finding the existence of infinite series of symmetries and conservation laws [1]. Recently, its singular analysis and Lax pair were given by Kalkanli, Sakovich and Yurdusen, using the Painlevé test and prolongation technique [3]. More recently, a series of exact wave solutions, including the solitary wave, rational, triangular periodic, Jacobi, and Weierstrass dou-