Nonlocal Modified Korteweg-de Vries Equation Research Articles

General soliton solutions for the complex reverse space-time nonlocal mKdV equation on a finite background

Three kinds of Darboux transformations are constructed by means of the loop group method for the complex reverse space-time (RST) nonlocal modified Korteweg–de Vries equation, which are different from that for the PT symmetric (reverse space) and reverse time nonlocal models. The N-periodic, the N-soliton, and the N-breather-like solutions, which are, respectively, associated with real, pure imaginary, and general complex eigenvalues on a finite background are presented in compact determinant forms. Some typical localized wave patterns such as the doubly periodic lattice-like wave, the asymmetric double-peak breather-like wave, and the solitons on singly or doubly periodic waves are graphically shown. The essential differences and links between the complex RST nonlocal equations and their local or PT symmetric nonlocal counterparts are revealed through these explicit solutions and the solving process.

Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations

In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg–de Vries equation. Based on general theory, Lie point symmetries are obtained and used to reduce these equations to nonlocal and local ordinary differential equations, separately; namely, one symmetry may allow reductions to both nonlocal and local equations, depending on how the invariant variables are chosen. For the nonlocal modified Korteweg–de Vries equation, analogously to the local situation, all reduced local equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations.

Inverse scattering transform for the complex reverse space-time nonlocal modified Korteweg-de Vries equation with nonzero boundary conditions and constant phase shift.

Recently, the complex reverse space-time (RST) nonlocal modified Korteweg-de Vries equation (mKdV) was introduced and shown to be an integrable infinite-dimensional dynamical system. The inverse scattering transform (IST) for zero boundary condition was studied by Ablowitz and Musslimani in 2016. In this paper, the IST for the complex RST nonlocal mKdV equation with nonzero boundary conditions at infinity is presented. The direct and inverse scattering problems are analyzed. The method to carry out the inverse problem employs two Riemann surfaces associated with square root branch points in the eigenfunctions/scattering data, which is more complicated than that for the zero boundary condition. Four cases are considered, special soliton solutions are discussed, and an explicit 1-soliton and two 2-soliton are found for three cases. In another case, there are no solitons.

Long-Time Asymptotics for the Nonlocal MKdV Equation**Supported by National Science Foundation of China under Grant Nos. 11671095, 51879045, National Science Foundation of China under Grant No. 11501365, and Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant No. 15YF1408100, Shanghai Youth Teacher Assistance Program No. ZZslg15056

In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation (nonlocal mKdV) qt(x, t) + qxxx(x,t) −6q(x, t)q(−x, −t)qx (x, t) = 0, which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal mKdV equation. In contrast with the classical mKdV equation, we find some new and different results on long-time asymptotics for the nonlocal mKdV equation and some additional assumptions about the scattering data are made in our main results.

New doubly-periodic solutions for the new integrable non local modified KdV equation

In this paper, new doubly-periodic solutions will be found for a new integrable nonlocal modified Korteweg-de Vries equation (mKdV). We adopt a technique using a travelling wave transformation which results in different solutions represented in Jacobi elliptic functions.

Soliton solutions of an integrable nonlocal modified Korteweg–de Vries equation through inverse scattering transform

It is well known that the nonlinear Schrödinger (NLS) equation is a very important integrable equation. Ablowitz and Musslimani introduced and investigated an integrable nonlocal NLS equation through inverse scattering transform. Very recently, we proposed an integrable nonlocal modified Korteweg–de Vries equation (mKdV) which can also be found in the papers of Ablowitz and Musslimani. We have constructed the Darboux transformation and soliton solutions for the nonlocal mKdV equation. In this paper, we will investigate further the nonlocal mKdV equation. We will give its exact solutions including soliton and breather through inverse scattering transformation. These solutions have some new properties, which are different from the ones of the mKdV equation.

A new integrable nonlocal modified KdV equation: Abundant solutions with distinct physical structures

In this work we study a new integrable nonlocal modified Korteweg–de Vries equation (mKdV) which arises from a reduction of the AKNS scattering problem. We use a variety of distinct techniques to determine abundant solutions with distinct physical structures. We show that this nonlocal equation possesses a family of traveling solitary wave solutions that include solitons, kinks, periodic and singular solutions.

On a nonlocal modified Korteweg-de Vries equation: Integrability, Darboux transformation and soliton solutions

Very recently, Ablowitz and Musslimani introduced a new integrable nonlocal nonlinear Schrödinger equation. In this paper, we investigate an integrable nonlocal modified Korteweg-de Vries equation (mKdV) which can be derived from the well-known AKNS system. We construct the Darboux transformation for the nonlocal mKdV equation. Using the Darboux transformation, we obtain its different kinds of exact solutions including soliton, kink, antikink, complexiton, rogue-wave solution, and nonlocalized solution with singularities. It is shown that these solutions possess new properties which are different from the ones for mKdV equation.