Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations

TL;DR

This paper extends Lie's symmetry methods to complex differential equations, showing how their linearization can be used to linearize systems of partial differential equations through complex and real transformations.

Contribution

It develops invariant criteria for linearizing complex ODEs and applies these to systems of PDEs, providing a novel approach to solving nonlinear PDEs.

Findings

Complex Lie symmetries enable linearization of PDE systems.

Explicit criteria for linearization are derived.

Transformations can be used to solve nonlinear PDEs explicitly.

Abstract

The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations implies the linearizability of systems of partial differential equations corresponding to those complex ordinary differential equations. The invertible complex transformations can be used to obtain invertible real transformations that map a system of nonlinear partial differential equations into a system of linear partial differential equation. Explicit invariant criteria are given that provide procedures for writing down the solutions of the linearized equations. A few non-trivial examples are mentioned.