Discrete mKdV Equation Research Articles

On the Ablowitz–Ladik equations with self-consistent sources

Darboux transformations and explicit solutions to Ablowitz–Ladik (AL) equations with self-consistent sources (ALESCS) are studied. Based on the Darboux transformation (DT) for the AL problem, we construct three types of non-auto-Bäcklund transformations connected with AL systems with different numbers of sources. The degenerate cases of DT and their applications to the reduced systems of ALESCS, for instance, discrete nonlinear Schrödinger with self-consistent sources (D-NLSSCS) and discrete mKdV equation with self-consistent sources (D-mKdVSCS), are discussed. Many types of solutions of ALESCS, D-NLSSCS and D-mKdVSCS, including solitons, positons, negatons can be derived from DTs and their degenerate cases.

Constructing the exact solutions of the (2+1)-dimensional Hybrid-Lattice and discrete mKdV equation

The paper gives a method for combining the auxiliary equation with hyperbolic function by function transition. And then the method is applied to construct the new solitary wave solutions and the trigonomical function wave solutions to the (2+1)-dimensional Hybrid-Lattice and discrete mKdV equation with the help of the symbolic computation system Mathematica.

Rational formal solutions of differential-difference equations

Abstract In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansatz. With the help of symbolic computation Maple , we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.

DISCRETE INTEGRABLE SYSTEMS ASSOCIATED WITH THE UNITARY MATRIX MODEL

The orthogonal polynomials on the unit circle associated with the unitary matrix model have various interesting properties, and have been studied in connection with different applications, such as double scaling, Riemann–Hilbert problems and integrable Fredholm operators. In this paper, we study the orthogonal polynomials on the unit circle with the weight function [Formula: see text]. We use the orthogonality of the polynomials to show that the orthogonal polynomials on the unit circle satisfy the linear problems associated with the discrete PainlevéII hierarchy, alternate discrete PainlevéII hierarchy and the discrete MKdV hierarchy. Thus the isomonodromy deformation method can be used to study the unitary matrix model. Typically, we focus on the orthogonal polynomials with the weight function exp {s(z+z-1)}. The recursion formula z(pn+vnpn-1)=pn+1+unpn (derived from Szegö's equation) for the orthogonal polynomials pn(z,s) is proved to be equivalent to the discrete AKNS-ZS system which is the base equation in the linear problem for the discrete equations. The variable xn=pn(0,s) satisfies the discrete PainlevéII equation and the discrete MKdV equation; un, vn satisfy the alternate discrete PainlevéII equation; 1/un-1=-xn-1/xn satisfies the PainlevéIII equation; and [Formula: see text] satisfies the PainlevéV equation. Also, we discuss the continuum limits of the discrete equations by using the double scaling method. Further, we present a procedure for finding the linear equations for the orthogonal polynomials and the consistency conditions of the linear equations by using the orthogonality of the polynomials.

1,2-rational N-soliton solutions for difference–differential equations related to the Volterra equation

We present a new 1-rational N-soliton solution for the Volterra equation. We also present an additional note on 1,2-rational N-soliton solutions for already-known equations related to the Volterra equation. Next, we present a new representation of an already-known 1-rational N-soliton solution for the discrete MKdV equation and a new 1-rational N-soliton solution for the discrete KdV equation. Lastly, we present 1-rational N-soliton solutions for the difference–differential equations related to the discrete MKdV equation.

Integrable discretization of the modified KdV equation and applications

An integrable discretization of the modified Korteweg-de Vries (mKdV) equation is considered. The mKdV algorithm is designed in terms of the resulting discrete mKdV equation. It is shown that a class of algebraic equations can be solved numerically in terms of the algorithm.

Multiple Shock Wave Solution for New Highly Nonlinear Difference-Differential Equation Related to the Discrete MKdV Equation

Eight types of highly nonlinear difference-differential equations are classified into three doublets and two exceptions. They consist of already-presented seven difference-differential equations and a new difference-differential equation. By reducing the new equation to seven bilinear equations, its multiple shock wave solution is obtained.

Lax representations and integrable time discretizations of the DDKdV, DDmKdV, and DDHOKdV

From a proper 2 × 2 discrete isospectral problem, a new differential-difference integrable equation in Lax sense is proposed by a discrete zero curvature equation. The DDKdV (differential-difference DdV equation) proposed by Ohta and Hirota and DDCDGKS (DD Caudrey-Dodd-Gibbon-Kotera-Sawada equation) are rederived. Some other new discrete KdV equations, discrete mKdV equations and discrete high order KdV equations which converge to the corresponding continuous soliton equations in the continuum limit are obtained. Integrable time discretizations of the DDKdV, DDmKdV (differential-difference mKdV equation) and DDHOKdV (differential-difference high order KdV equations) are given.

Miura Transformations between Sokolov-Shabat's Equation and the Discrete MKdV Equation

Miura Transformations between Sokolov-Shabat's Equation and the Discrete MKdV Equation

Bilinearization of discrete soliton equations and singularity confinement

The singularity confinement method is applied to the systematic derivation of the bilinear equations for discrete soliton equations. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed.

A comparison of two discrete mKdV equations

We consider here two discrete versions of the modified KdV equation. In one case, some solitary wave solutions, Bäcklund transformations and integrals of motion are known. In the other one, only solitary wave solutions were given, and we supply the corresponding results for this equation. We also derive the integrability of the second equation and give a transformation which links the two models.

Integrable Dynamics of Discrete Surfaces II

The discrete surfaces (triangulate surfaces) in 3-dimensional space are mapped to the sphere S 2 and differential-difference equations are obtained to describe the motion of the discrete surfaces on the S 2 . These equations contain integrable dynamics such as the discrete nonlinear Schrödinger equation and the discrete complex mKdV equation. Relations between the motion of discrete surfaces and nonlinear integrable differential-difference equations are discussed.

Decoupling of the N-soliton solution for a new discrete MKdV equation

We prove that the N-soliton solution for a new discrete MKdV equation can be decoupled into N components. These components satisfy the interacting new discrete MKdV equation, and approach single soliton solutions as t→∓∞. We present concrete expressions of soliton components in the case of N = 2, and illustrate the results.