Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations

Linyu Peng

doi:10.3390/sym11070884

Abstract

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.

Highlights

Symmetry has proved to be fundamentally important in understanding the solutions of differential equations
It reveals the integrability of partial differential equations (PDEs); for instance, the Ablowitz–Ramani–Segur conjecture stated that every ordinary differential equation (ODE) obtained by an exact reduction of an integrable evolution equation solvable by inverse scattering transforms is of the P-type, i.e., ODEs without movable critical points [6]
We find that one may reduce a nonlocal differential equation to both nonlocal and local ODEs by choosing the invariant variables in different ways

Summary

Introduction

Symmetry has proved to be fundamentally important in understanding the solutions of differential equations (see, e.g., [1,2,3,4,5]). Integrable nonlocal systems have recently received a great amount of attention with many newly-proposed models (e.g., the nonlocal vector NLS equation [13], a multi-dimensional extension of the nonlocal NLS equation [10], the nonlocal sine-Gordon equation, the nonlinear derivative NLS equation and related systems [9], the nonlocal mKdV equation [8], Alice–Bob physics [11], and the nonlocal Sasa–Satsuma equation [12], to mention only a few) Solutions of these systems have been explored by many scholars; see, for example, [8,12,14,15,16,17,18,19].

The Linearized Symmetry Condition for Nonlocal Differential Equations

The Nonlocal NLS Equation

Lie Point Symmetries

Symmetry Reductions

The Nonlocal mKdV Equation

Conclusions