# Generalized symmetry reduction of nonlinear differential equations

## TL;DR

This paper develops a generalized symmetry method for reducing nonlinear PDEs, enabling the construction of solutions beyond classical Lie approaches, including solutions depending on arbitrary functions.

## Contribution

It introduces a generalized symmetry reduction technique using Lie-Bäcklund operators, applicable to various nonlinear PDEs, and explores the relationship between symmetry algebra size and non-invariant solutions.

## Key findings

- Constructed ansatzes reduce PDEs to ODE systems.
- Found solutions depending on arbitrary functions.
- Linked symmetry algebra dimension to solution types.

## Abstract

We study the application of generalized symmetry for reducing nonlinear partial differential equations. We construct the ansatzes for dependent variable $u$ which reduce the scalar partial differential equation with two independent variables to systems of ordinary differential equations. The operators of Lie-B\"acklund symmetry of the second order ordinary differential equation are used. We apply the method to nonlinear evolutionary equations and find solutions which cannot be obtained in the framework of classical Lie approach. The method is also applicable to partial differential equations which are not restricted to evolution type ones. We construct the solution of nonlinear hyperbolic equation depending on an arbitrary smooth function on one variable. We study also the correlation between the dimension of symmetry Lie algebra and possibility of constructing non-invariant solutions to the equation under study.