Quasi-algorithmical construction of reciprocal transformations

TL;DR

This paper explores a quasi-algorithmical method for constructing reciprocal transformations, which are used to simplify or linearize nonlinear PDEs and identify equivalences among seemingly different equations in physics and mathematics.

Contribution

It introduces a novel quasi-algorithmical approach to construct reciprocal transformations, aiding in the classification and simplification of nonlinear PDEs.

Findings

Reciprocal transformations can reveal equivalences between different PDEs.

The proposed method helps identify underlying common problems in seemingly unrelated equations.

Applications include simplifying complex equations in physics and mathematics.

Abstract

Reciprocal transformations mix the role of the dependent and independent variables to achieve simpler versions or even linearized versions of nonlinear PDEs. These transformations help in the identification of a plethora of PDEs available in the Physics and Mathematics literature. Two different equations, although seemingly unrelated, happen to be equivalent versions of a same equation after a reciprocal transformation. In this way, the big number of integrable equations could be greatly diminished by establishing a method to discern which equations are disguised versions of a common underlying problem. Then, a question arises: Is there a way to identify different versions of an underlying common nonlinear problem? Other useful applications of reciprocal transformations are subsequently discussed and illustrated with examples.