Linearizability for third order evolution equations

TL;DR

This paper investigates conditions under which third order evolution equations can be linearized, identifying a class of such equations with infinite-dimensional symmetry groups and deriving explicit linearizing transformations.

Contribution

It introduces criteria for linearity testing, characterizes a broad class of linearizable equations, and connects these to well-known integrable equations like KdV and mKdV.

Findings

Criteria for testing linearity of third order evolution equations

Identification of a class of linearizable equations with arbitrary functions

Explicit linearizing transformations and connections to KdV/mKdV equations

Abstract

The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite-dimensional symmetry group. Linearizing transformations for this class are found using symmetry structure and local conservation laws. A number of special cases as examples are discussed. Their transformation to equations within the same class by differential substitutions and connection with KdV and mKdV equations are also reviewed in this framework.