Related papers: On real hyperelliptic solutions of focusing modifi…
Generalization of the modified KdV equation to a multi-component system, that is expressed by $(\partial u_i)/(\partial t) + 6 (\sum_{j,k=0}^{M-1} C_{jk} u_j u_k) (\partial u_i)/(\partial x) + (\partial^3 u_{i})/(\partial x^3) = 0, i=0, 1,…
A simple and general approach for calculating the elliptic finite-gap solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is based on the use of the finite-gap equations and the general representation of these…
We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing mKdV equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the $\tau$…
In the present paper, a hierarchy of the mKdV equation is integrated by the methods of algebraic geometry. The mKdV hierarchy in question arises on coadjoint orbits in the loop algebra of $\mathfrak{sl}(2)$, and employs a family of…
In this work, we primarily focus on the two-phase solutions and their stability to the focusing mKdV equation. By employing the algebro-geometric approach in combination with an effective integration method, we construct explicit two-phase…
This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which…
We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $M^{2,p}_{s}(\mathbb{R})$ for $s \ge \frac14$ and $2\leq p < \infty$. For…
We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense (in the spirit of [18]) and which remain close to multi-solitons. We show that these solutions are necessarily pure…
In this paper, the complex version KdV equation is discussed. The corresponding coupled equations is a integrable system in the sense of the bi-Hamiltonian structure, so the complex version KdV equation is integrable. A new spectral form is…
We consider the one dimensional periodic complex valued mKdV, which corresponds to the first equation above cubic NLS in the associated integrable hierarchy. Our main result is the construction of a sequence of invariant measures supported…
We study the plus and minus type discrete mKdV equation. Some different symmetry conditions associated with two Lax pairs are introduced to derive the matrix Riemann-Hilbert problem with zero. By virtue of regularization of the…
A new matrix modified Korteweg-de Vries (mmKdV) equation with a $p\times q$ complex-valued potential matrix function is first studied via Riemann-Hilbert approach, which can be reduced to the well-known coupled modified Korteweg-de Vries…
Let $X$be a complex hyperelliptic curve of genus two equipped with the canonical metric $ds^2$. We study mean field equations on complex hyperelliptic curves and show that the Gaussian curvature function of $(X,ds^2)$ determines an explicit…
Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the…
In this paper, a generalized variable-coefficient KdV equation (vcKdV) arising in fluid mechanics, plasma physics and ocean dynamics is investigated by using symmetry group analysis. Two basic generators are determined, and for every…
Using the fact that Miura transformation can be expressed in the form of gauge transformation connecting the KdV and mKdV equations, we discuss the derivation of the B\"acklund transformation and its Miura-gauge transformation connecting…
We consider equations in the modified KdV (mKdV) hierarchy and make use of the Miura transformation to construct expressions for their Lax pair. We derive a Lagrangian-based approach to study the bi-Hamiltonian structure of the mKdV…
In this paper, we prove that the nonlocal complex modified Korteweg-de Vries (mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity, 29, 915-946 (2016)] is gauge equivalent to a spin-like model. From the gauge equivalence, one…
In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schr\"odinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather…
For a generalized super KdV equation, three Darboux transformations and the corresponding B\"acklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax…