Related papers: On real hyperelliptic solutions of focusing modifi…
Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a…
We construct dynamics for the defocusing real-valued (Miura) mKdV equation on the real line with initial data distributed according to Gibbs measure. We also prove that Gibbs measure is invariant under these dynamics. On the way, we provide…
We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.
Soliton Solutions of Korteweg-de Vries (KdV) were constructed for given degenerate curves $y^2 = (x-c)P(x)^2$ in terms of hyperelliptic sigma functions and explicit Abelian integrals. Connection between sigma functions and tau function were…
Let $d$ be a positive integer, $\mathbb K$ an algebraically closed field of characteristic 0 and $ X$ an elliptic curve defined over K. We study the hyperelliptic curves equipped with a projection over $ X$, such that the natural image of $…
The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of integrable…
We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations.…
In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions…
In this paper, we consider matrix Schr\"odinger equation, dynamical Schr\"odinger equation and matrix KdV. We construct their explicit solutions using our GBDT version of B\"acklund--Darboux transformation and square roots of the…
We construct a deformation quantized version (ncKdV) of the KdV equation which possesses an infinite set of conserved densities. Solutions of the ncKdV are obtained from solutions of the KdV equation via a kind of Seiberg-Witten map. The…
We study the nonlocal modified Korteweg-de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz-Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then…
We prove that the modified Korteweg- de Vries equation (mKdV) equation is unconditionally well-posed in $H^s(\mathbb R)$ for $s> \frac 13$. Our method of proof combines the improvement of the energy method introduced recently by the first…
We study the recently-proposed hyperbolic approximation of the Korteweg-de Vries equation (KdV). We show that this approximation, which we call KdVH, possesses a rich variety of solutions, including solitary wave solutions that approximate…
Nonlinear non-Abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Baecklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified…
Multi-kink solutions of the defocusing, modified Korteweg-de Vries equation (mKdV) found by Grosse are shown to be globally $H^1$-stable. Stability in the one-kink case was previously established by Zhidkov, and Merle-Vega. The proof uses…
This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg-de Vries equation, the Kuramoto-Sivashinsky equation, the generalized Korteweg-de Vries-Kuramoto-Sivashinski equation and the non…
We introduce a parametric coupled KdV system which contains, for particular values of the parameter, the complex extension of the KdV equation and one of the Hirota-Satsuma integrable systems. We obtain a generalized Gardner transformation…
We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{k+1})=0$, where $k\geq5$ is an integer number and $\mu=\pm1$. In the focusing case ($\mu=1$), we show that if the initial data $u_0$…
In order to find higher dimensional integrable models, we study differential equations of hyperelliptic $\wp$ functions up to genus four. For genus two, differential equations of hyperelliptic $\wp$ functions can be written in the Hirota…
In this paper we show some exact solutions for the general fifth order KdV equation. These solutions are obtained by the extended tanh method.