Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

TL;DR

This paper introduces new methods for generating solutions to nonlinear PDEs using nonlocal symmetries, linearizations, and superposition formulas, enabling the construction of chains of exact solutions.

Contribution

It develops novel nonlocal symmetry-based solution formulas for nonlinear heat and Burgers equations, and applies linearization techniques to derive superposition principles for complex PDEs.

Findings

New solution formulas for nonlinear heat and Burgers equations

Construction of chains of exact solutions using nonlocal symmetries

Application of Legendre transformations for solution generation

Abstract

We have constructed new formulae for generation of solutions for the nonlinear heat equation and for the Burgers equation that are based on linearizing nonlocal transformations and on nonlocal symmetries of linear equations. Found nonlocal symmetries and formulae of nonlocal nonlinear superposition of solutions of these equations were used then for construction of chains of exact solutions. Linearization by means of the Legendre transformations of a second-order PDE with three independent variables allowed to obtain nonlocal superposition formulae for solutions of this equation, and to generate new solutions from group invariant solutions of a linear equation.