Reciprocal transformations and their role in the integrability and classification of PDEs

TL;DR

This paper explores reciprocal transformations that interchange variables in nonlinear PDEs, aiding in their classification and revealing hidden equivalences among seemingly different equations to better understand their integrability.

Contribution

It introduces a method to identify when different PDEs are equivalent under reciprocal transformations, simplifying the classification of integrable equations.

Findings

Reciprocal transformations can linearize certain nonlinear PDEs.

Many equations in literature are equivalent under reciprocal transformations.

A framework is proposed to recognize disguised versions of the same PDE.

Abstract

Reciprocal transformations mix the role of the dependent and independent variables of (nonlinear partial) differential equations to achieve simpler versions or even linearized versions of them. These transformations help in the identification of a plethora of partial differential equations that are spread out in the physics and mathematics literature. Two different initial equations, although seemingly unrelated at first, could be the same equation after a reciprocal transformation. In this way, the big number of integrable equations that are spread out in the literature could be greatly diminished by establishing a method to discern which equations are disguised versions of a same, common underlying equation. Then, a question arises: Is there a way to identify different differential equations that are two different versions of a same equation in disguise?