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...

In this article, we study and investigate the analytical solutions of space-time nonlinear fractional modified KDV-Zakharov-Kuznetsov (mKDV-ZK) equation. We have got new exact mKDV-ZK equation by using first integral method; found types hyperbolic trigonometric symbolic computation.

Collocation method using quintic B-splines finite element have been developed for solving numerically the HirotaSatsuma coupled MKdV equation. Accuracy of the proposed method is shown numerically by calculating conservation laws, 2 L and  L norms on studying of a soliton solution. It is shown that the collocation scheme for solutions of the MKdV equation gives rise to smaller errors and is qui...

We show that an interacting double soliton solution to the perturbed mKdV equation (1.1). is close in H2 to a double soliton following an effective dynamics obtained as Hamilton’s equations for the restriction of the mKdV Hamiltonian to the submanifold of solitons. The interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to sol...

We investigate the mKdV hierarchy with integral type of source (mKdVHWS), which consist of the reduced AKNS eigenvalue problem with r = q and the mKdV hierarchy with extra term of the integration of square eigenfunction. First we propose a method to find the explicit evolution equation for eigenfunction of the auxiliary linear problems of the mKdVHWS. Then we determine the evolution equations o...

In this work, we established exact solutions for the combined KdV-MKdV equation. By constructing four new types of Jacobi elliptic functions solutions, the Jacobi elliptic functions expansion method will be extend. With the aid of symbolic computation system mathematica, obtain some new exact periodic solutions of nonlinear combined KdV-MKdV equation , and these solutions are degenerated to sol...

Rogue periodic waves stand for rogue waves on the periodic background. Two families of traveling periodic waves of the modified Korteweg–de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn. By using one-fold and twofold Darboux transformations, we construct explicitly the rogue periodic waves of the mKdV equation. Since the dn-periodic wave i...

The matrix 2x2 spectral differential equation of the second order is considered on x in (−∞, +∞). We establish elementary Darboux transformations covariance of the problem and analyze its combinations. We select a second covariant equation to form Lax pair of a coupled KdV-MKdV system. The sequence of the elementary Darboux transformations of the zero-potential seed produce two-parameter soluti...

Abstract We connect certain continuous motions of discrete planar curves resulting in semi-discrete potential Korteweg–de Vries (mKdV) equation with Darboux transformations smooth curves. In doing so, we define infinitesimal that include the aforementioned motions, and also give an alternate geometric interpretation for establishing mKdV equation.

For N ≥ 3 there are SN and DN actions on the space of solutions of the first nontrivial equation in the SL(N) MKdV hierarchy, generalizing the two Z2 actions on the space of solutions of the standard MKdV equation. These actions survive scaling reduction, and give rise to transformation groups for certain (systems of) ODEs, including the second, fourth and fifth Painlevé equations. Given a solu...