Related papers: Discrete mKdV equation via Darboux transformation
The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac…
New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem.…
The main aim of this paper is to investigate Darboux rectifying curves on a smooth surface immersed in the Euclidean space. First, we discuss the component of the position vector of a Darboux rectifying curve on a smooth immersed surface…
The Darboux transformation operator technique in differential and integral forms is applied to the generalized Schrodinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. Intertwining…
In this paper, we apply the moving plane method to some degenerate elliptic equations to get a Liouville type theorem. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic…
We use the singular manifold method to obtain the Lax pair, Darboux transformations and soliton solutions for a (2+1) dimensional integrable equation.
In this paper we present a set of results on the integration and on the symmetries of the lattice potential Korteweg-de Vries (lpKdV) equation. Using its associated spectral problem we construct the soliton solutions and the Lax technique…
In this article, we will report the recent developments on Lax pairs and Darboux transformations for Euler equations of inviscid fluids.
We construct a discrete version of the plane wave solution to a discrete Dirac-K\"{a}hler equation in the Joyce form. A geometric discretisation scheme based on both forward and backward difference operators is used. The conditions under…
We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow…
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…
Singular Darboux transformations, in contrast to the conventional ones, have a singular matrix as a coefficient before the derivative. We incorporated such transformations into a chain of conventional transformations and presented…
The Korteweg-de Vries (KdV) equation is of fundamental importance in a wide range of subjects with generalization to multi-component systems relevant for multi-species fluids and cold atomic mixtures. We present a general framework in which…
The exactly solvable scalar-tensor potential of the four-component Dirac equation has been obtained by the Darboux transformation method. The constructed potential has been interpreted in terms of nucleon-nucleon and Schwinger interactions…
We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most…
In this paper, we obtain a uniform Darboux transformation for multi-component coupled NLS equations, which can be reduced to all previous presented Darboux transformation. As a direct application, we derive the single dark soliton and…
In this paper we study weak continuity of the dynamical systems for the KdV equation in H^{-3/4}(R) and the modified KdV equation in H^{1/4}(R). This topic should have significant applications in the study of other properties of these…
Asymptotic stability is with no doubts an essential property to be studied for any system. This analysis often becomes very difficult for coupled systems and even harder when different timescales appear. The singular perturbation method…
In this study we examine the energy transfer mechanism during the nonlinear stage of the Modulational Instability (MI) in the modified Korteweg-de Vries equation. The particularity of this study consists in considering the problem…
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…