# Nonlocal symmetry, Darboux transformation and soliton-cnoidal wave interaction solution for the shallow water wave equation

## TL;DR

This paper derives nonlocal symmetries and Darboux transformations for the AKNS-type shallow water wave equation, enabling explicit solutions including soliton-cnoidal wave interactions, with detailed analysis and graphical illustrations.

## Contribution

It introduces a method to localize nonlocal symmetries and constructs Darboux transformations for the AKNS-type SWW equation, leading to new explicit interaction solutions.

## Key findings

- Derived nonlocal symmetries from the Lax pair.
- Constructed Darboux transformation for the extended system.
- Obtained analytical soliton-cnoidal wave interaction solutions.

## Abstract

In classical shallow water wave (SWW) theory, there exist two integrable one-dimensional SWW equation [Hirota-Satsuma (HS) type and Ablowitz-Kaup-Newell-Segur (AKNS) type] in the Boussinesq approximation. In this paper, we mainly focus on the integrable SWW equation of AKNS type. The nonlocal symmetry in form of square spectral function is derived starting from its Lax pair. Infinitely many nonlocal symmetries are presented by introducing the arbitrary spectrum parameter. These nonlocal symmetries can be localized and the SWW equation is extended to enlarged system with auxiliary dependent variables. Then Darboux transformation for the prolonged system is found by solving the initial value problem. Similarity reductions related to the nonlocal symmetry and explicit group invariant solutions are obtained. It is shown that the soliton-cnoidal wave interaction solution, which represents soliton lying on a cnoidal periodic wave background, can be obtained analytically. Interesting characteristics of the interaction solution between soliton and cnoidal periodic wave are displayed graphically.