# Lie-B\"acklund symmetry and non-invariant solutions of nonlinear evolution equations

## TL;DR

This paper introduces a method using Lie-Bäcklund symmetry to reduce nonlinear diffusion PDEs to ODEs, enabling the discovery of new solutions beyond classical Lie symmetry approaches.

## Contribution

The authors develop a novel approach employing Lie-Bäcklund symmetry operators to find non-invariant solutions of nonlinear evolution equations, expanding solution techniques beyond traditional methods.

## Key findings

- Constructed solutions for nonlinear diffusion equations with various Lie group invariances.
- Demonstrated the method's ability to find solutions unattainable by classical Lie symmetry.
- Provided ansatzes reducing PDEs to ODE systems using third-order Lie-Bäcklund operators.

## Abstract

We study the symmetry reduction of nonlinear partial differential equations which are used for describing diffusion processes in nonhomogeneous medium. We find ansatzes reducing partial differential equations to systems of ordinary differential equations. The ansatzes are constructed by using operators of Lie-B\"acklund symmetry of third order ordinary differential equation. The method gives the possibility to find solutions which cannot be obtained by virtue of classical Lie method. Such solutions have been constructed for nonlinear diffusion equations which are invariant with respect to one-parameter, two-parameter and three-parameter Lie groups of point transformations.