Similarity transformations and linearization for a family of dispersionless integrable PDEs

TL;DR

This paper uses Lie point symmetries to transform a family of integrable PDEs into linearizable second-order equations, revealing their maximal symmetry and simplifying their analysis.

Contribution

It demonstrates that all equations in the family can be reduced to linear second-order equations via similarity transformations derived from Lie symmetries.

Findings

All PDEs are reducible to linear second-order equations.

The reduced equations are maximally symmetric.

The method applies to PDEs integrable by hyperbolic reductions.

Abstract

We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlev\'{e} transcendents. The main results of this study is that from the application of the similarity transformations provided by the Lie point symmetries all the members of the family of the partial differential equations are reduced to second-order differential equations which are maximal symmetric and can be linearized.