We present two extensions of Wilson’s explanation of the Miura map from MKdV to KdV. In the first we explain the map of Svinolupov et.al. from a certain UrKdV-like equation to KdV, and in the second we explain Konopelchenko’s map from the modified KP equation to KP. In the course of the latter we introduce an “UrKP” system, with an infinite dimensional symmetry, providing us with a systematic m...

The aim of this work is to study asymptotically and numerically the interaction solitons with an external forcing a variable speed using forced modified Korteweg–de Vries equation (mKdV). We show that asymptotic predictions agree well numerical solutions for forcings constant linear speed. Regarding speed, we find regimes in which are trapped at its amplitude increases or decreases time dependi...

Abstract We study the well-posedness of complex-valued modified Korteweg-de Vries equation (mKdV) on circle at low regularity. In our previous work (2021), we introduced second renormalized mKdV equation, based conservation momentum, which proposed as correct model to outside $$H^\frac{1}{2}({\mathbb {T}})$$ <mml:mi...

The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to nonconvex flux is studied in the framework modified Korteweg-de Vries (mKdV) equation, a canonical model for nonlinear internal gravity wave propagation stratified fluids. principal feature that both and large-scale flow -- rarefaction or shock (undular bore) are described by same hydrody...

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. In this scenario the numerical schemes which preserve the discrete invariants related...

The nolinear hydrodynamic equations of the surface of a liquid drop are shown to be directly connected to Korteweg de Vries (KdV, MKdV) systems, giving traveling solutions that are cnoidal waves. They generate multiscale patterns ranging from small harmonic oscillations (linearized model), to nonlinear oscillations, up through solitary waves. These non-axis-symmetric localized shapes are also d...

The Jacobin doubly periodic wave solution, the Weierstrass elliptic function solution, the bell-type solitary wave solution, the kink-type solitary wave solution, the algebraic solitary wave solution, and the triangular solution of a generalized Korteweg-de Vries-modified Korteweg-de Vries equation (GKdV-mKdV) with higher-order nonlinear terms are obtained by a generalized subsidiary ordinary d...

but it will be clear immediately to the reader with some experience in the field, that the method extends naturally and easily to the general class of wave equations solvable by the inverse scattering method, such as the KdV, nonlinear Schrödinger (NLS), and Boussinesq equations, etc., and also to "integrable" ordinary differential equations such as the Painlevé transcendents. As described, for...

A numerical behavior of a KdV combined mKdV equation is obtained using the Melnikov method. Melnikov method is proved to be elegant, and successful alternative to characterizing the complex dynamics of multi-stable oscillators. Based on the Melnikov theory we present the homoclinic and heteroclinic orbits in the unperturbed system. It is show us whether the system is chaotic or not. In this wor...

A method is proposed to construct closed-form solutions of nonlinear differential-difference equations. For the variety of nonlinearities, this method only deals with such equations which are written in polynomials in function and its derivative. Some closed-form solutions of Hybrid lattice, Discrete mKdV lattice, and modified Volterra lattice are obtained by using the proposed method. The trav...