# B\"acklund Transformation and Quasi-Integrable Deformation of Mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models

## TL;DR

This paper investigates a generalized nonlinear PDE derived from mixed Fermi-Pasta-Ulam and Frenkel-Kontorova models, deriving its Bäcklund transformation and exploring its quasi-integrable deformation to understand non-linear dislocation waves.

## Contribution

It introduces a new continuum PDE model with Bäcklund transformation and analyzes its quasi-integrable deformation, extending the understanding of non-linear lattice dynamics.

## Key findings

- Derived Bäcklund transformation in Riccati form.
- Established quasi-integrable deformation of the model.
- Connected the PDE to physical dislocation wave phenomena.

## Abstract

In this paper we study a non-linear partial differential equation (PDE), proposed by N. Kudryashov [arXiv:1611.06813v1[nlin.SI]], using continuum limit approximation of mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models. This generalized semi-discrete equation can be considered as a model for the description of non-linear dislocation waves in crystal lattice and the corresponding continuous system can be called mixed generalized potential KdV and sine-Gordon equation. We obtain the B\"acklund transformation of this equation in Riccati form in inverse method. We further study the quasi-integrable deformation of this model.