Nonlocal Modified Korteweg-de Vries Research Articles

New solutions of coupled nonlocal NLS and coupled nonlocal mKdV equations

We provide several novel solutions of the coupled Ablowitz–Musslimani (AM) version of the nonlocal nonlinear Schrödinger (NLS) equation and the coupled nonlocal modified Korteweg–de Vries (mKdV) equation. In each case we compare and contrast the corresponding solutions of the relevant coupled local equations. Further, we provide new solutions of the coupled local NLS and coupled local mKdV equations which are not the solutions of the corresponding nonlocal equations. We also show that the nonlocal coupled (as well as uncoupled) mKdV equations have hidden Galilean invariance and admit novel solutions.

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.

New solutions of nonlocal NLS, mKdV and Hirota equations

In this paper, we provide several novel solutions of the Ablowitz–Musslimani and Yang’s versions of the nonlocal nonlinear Schrödinger (NLS) equation, nonlocal modified Korteweg–de Vries (mKdV) as well as nonlocal Hirota equations. In each case we compare and contrast with the corresponding solutions of the relevant local equation. In addition, we provide new solutions of the local NLS, local mKdV and local Hirota equations which are not the solutions of the corresponding nonlocal equations.

Coupled superposed solutions in nonlinear nonlocal equations

We obtain novel periodic as well as hyperbolic solutions of an Ablowitz–Musslimani variant of the coupled nonlocal, nonlinear Schrödinger equation (NLS) as well as a coupled nonlocal modified Korteweg–de Vries (mKdV) equation which can be re-expressed as a linear superposition of the sum or the difference of two hyperbolic or two periodic kink or pulse solutions. Besides, we also discuss some of the other solutions admitted by these coupled equations. These results demonstrate that the notion of the superposed solutions extends to the coupled nonlocal nonlinear equations as well.

Novel superposed kinklike and pulselike solutions for several nonlocal nonlinear equations

We show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant of the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal modified Korteweg de Vries (mKdV) equation, and the nonlocal Hirota equation, admit novel kinklike and pulselike superposed periodic solutions. Furthermore, we show that the nonlocal mKdV equation also admits the superposed (hyperbolic) kink–antikink solution. In addition, we show that while the nonlocal Ablowitz–Musslimani variant of the NLS admits complex parity-time reversal-invariant kink and pulse solutions, neither the local NLS nor the Yang variant of the nonlocal NLS admits such solutions. Finally, except for the Yang variant of the nonlocal NLS, we show that the other three nonlocal equations admit both the kink and pulse solutions in the same model.

On a Riemann–Hilbert problem for the focusing nonlocal mKdV equation with step-like initial data

We study the Cauchy problem for the integrable nonlocal modified Korteweg–deVries (mKdV) equation with a step-like initial data: q(x,0)=q0(x), where q0(x)=o(1) as x→−∞ and q0(x)=A+o(1), as x→∞, A>0. We construct the solution of the nonlocal mKdV equation via the solution of a 2 × 2 matrix Riemann–Hilbert problem in the complex plane. Further, The explicit form of the one-soliton solution in a special case is expressed in terms of the Riemann–Hilbert problem.

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.

Nonlocal modified KdV equations and their soliton solutions by Hirota Method

We study the nonlocal modified Korteweg–de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz–Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then by using the Ablowitz–Musslimani reduction formulas, we find one-, two-, and three-soliton solutions of nonlocal mKdV and nonlocal complex mKdV equations. The soliton solutions of these equations are of two types. We give one-soliton solutions of both types and present only first type of two- and three-soliton solutions. We illustrate our solutions by plotting their graphs for particular values of the parameters.

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.