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...

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 two-fold Darboux transformations of the travelling periodic waves, we construct new explicit solutions to the mKdV equation. Since t...

Solitary wave interaction for a higher-order modified Korteweg-de Vries (mKdV) equation is examined. The higher-order mKdV equation can be asymptotically transformed to the mKdV equation, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive the higher-order two-soliton solution and it is shown that the interaction is asymptotically elas...

We investigate bi–Hamiltonian structures and related mKdV hierarchy of solitonic equations generated by (semi) Riemannian metrics and curve flow of non–stretching curves. The corresponding nonholonomic tangent space geometry is defined by canonically induced nonlinear connections, Sasaki type metrics and linear connections. One yields couples of generalized sine–Gordon equations when the corres...

1 يوجشناد سانشراک ی ،هیذغت مولع دشرا ،نارهت یکشزپ مولع هاگشناد ،یسانش میژر و هیذغت هدکشناد ناریا 2 لوئسم هدنسیون : هورگ رایداتسا هیذغت هعماج ،ناریا ،نارهت یکشزپ مولع هاگشناد ،یسانش میژر و هیذغت هدکشناد ، یکینورتکلا تسپ : [email protected] 3 داتسا هورگ هیذغت هعماج هدکشناد، هیذغت میژرو هاگشناد،ینامرد مولع یکشزپ ،نارهت ناریا 4 يارتکد تملاس تاقیقحت یلم وتیتسنا ،ناهفصا یتشادهب ت...

The bi-Hamiltonian structure of the two known vector generalizations of the mKdV hierarchy of soliton equations is derived in a geometrical fashion from flows of nonstretching curves in Riemannian symmetric spaces G/SO(N). These spaces are exhausted by the Lie groups G = SO(N + 1), SU(N). The derivation of the bi-Hamiltonian structure uses a parallel frame and connection along the curve, tied t...

In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV) equations, namely, the KdV equation, the modified KdV (mKdV) equation, the potential KdV (pKdV) equation, the generalized KdV (gKdV) equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV ...