Related papers: Discrete mKdV equation via Darboux transformation
We systematically study Darboux-type transformations for the KdV and AKNS hierarchies and provide a complete account of their effects on hyperelliptic curves associated with algebro-geometric solutions of these hierarchies.
We study the Darboux transformation (DT) for Dirac equations with (1+1) potentials. Exact solutions for the adiabatic external field are constructed. The connection between the exactly soluble Dirac (1+1) potentials and the soliton…
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…
Using the Lie's infinitesimal method we establish that the Dirac equation in one variable is integrable by quadratures if the potential V(x) is a solution of one of the equations of the stationary mKdV hierarchy.
The Lax representation for the nonstationary Schr\"odinger equation with rather arbitrary potential is proposed. Some examples of the construction of exact solutions are given by means of Darboux Transformation method.
We present a unified operator-theoretic framework for constructing deterministic KdV soliton gases and step-type KdV solutions. Starting from Dyson's determinantal formula, we obtain a broad class of reflectionless solutions and describe…
We construct new quasi-exactly solvable one-dimensional potentials through Darboux transformations. Three directions are investigated: Reducible and two types of irreducible second-order transformations. The irreducible transformations of…
With the nonuniform media taken into account, the nonisospectral and variable-coefficient Korteweg-de Vries equation, which describes various physical situations such as fluid dynamics and plasma, is under investigation in this paper. With…
One way of constructing explicit expressions of solutions of integrable systems of Partial Differential Equations (PDEs) goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to…
A method is presented to obtain the change in the potential and in the relevant wavefunction of a linear system of ordinary differential equations containing a spectral parameter, when that linear system is perturbed and a finite number of…
Bilinear forms for some nonlinear partial difference equations(discrete soliton equations) are derived based on the results of singularity confinement. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete…
We consider here two discrete versions of the modified KdV equation. In one case, some solitary wave solutions, B\"acklund transformations and integrals of motion are known. In the other one, only solitary wave solutions were given, and we…
Under investigation in this paper is the nonisospectral and variable coefficients modified Kortweg-de Vries (vc-mKdV) equation, which manifests in diverse areas of physics such as fluid dynamics, ion acoustic solitons and plasma mechanics.…
The discrete non-commutative Darboux system of equations with self-consistent sources is constructed, utilizing both the vectorial fundamental (binary Darboux) transformation and the method of additional independent variables. Then the…
A new Darboux transformation is presented for the Hirota-Satsuma coupled KdV system. It is shown that this Darboux transformation can be constructed by means of two methods: Painlev\'{e} analysis and reduction of a binary Darboux…
Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides…
Rarefactive waves and dispersive shock waves are generated from the step-like initial data in many nonlinear evolution equations including the classical example of the Korteweg-de Vries (KdV) equation. When a solitary wave is injected on…
The Darboux transformation between ordinary differential equations is a 19th century technique that has seen wide use in quantum theory for producing exactly solvable potentials for the Schr\"odinger equation with specific spectral…
We present a new exact solution to the defocusing modified Korteweg-de Vries equation to describe the interaction of a dark soliton and a traveling periodic wave. The solution (which we refer to as to the dark breather) is obtained by using…
In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving…