# A nonlocal variable coefficient modified KdV equation derived from two-layer fluid system and its exact solutions

## TL;DR

This paper derives a nonlocal variable coefficient modified KdV equation from a two-layer fluid system using symmetry reduction and multiple scale methods, and finds various exact solutions including solitons and periodic waves.

## Contribution

It introduces a novel nonlocal VCmKdV equation from fluid dynamics and provides explicit exact solutions and an approximate solution for the original system.

## Key findings

- Exact solutions including elliptic periodic and solitary waves.
- Interaction solutions between solitons and periodic waves.
- An approximate solution for the original nonlocal system.

## Abstract

A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, then by applying multiple scale method to it a general nonlocal two place variable coefficient modified KdV (VCmKdV) equation with shifted space and delayed time reversal is derived. Various exact solutions of the VCmKdV equation, including elliptic periodic waves, solitary waves and interaction solutions between solitons and periodic waves are obtained and analyzed graphically. As an illustration, an approximate solution of the original nonlocal two-layer fluid system is also given.