# Symmetries and reductions of integrable nonlocal partial differential equations

## TL;DR

This paper extends symmetry analysis to nonlocal integrable PDEs, deriving reductions to local and nonlocal ODEs, and explores connections between nonlocal equations and differential-difference equations.

## Contribution

It introduces a symmetry-based reduction framework for nonlocal integrable equations and establishes links to differential-difference equations.

## Key findings

- Reduced nonlocal and local ODEs from nonlocal PDEs.
- All reduced local equations for the nonlocal mKdV are integrable.
- Connections between nonlocal PDEs and differential-difference equations are established.

## Abstract

In this paper, symmetry analysis is extended to study nonlocal differential equations, in particular two integrable nonlocal equations, the nonlocal nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries equation. Lie point symmetries are obtained based on a general theory 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. At the end, we also define complex transformations to connect nonlocal differential equations and differential-difference equations.