Unified approach to Miura, B\"acklund and Darboux transformations for nonlinear partial differential equations

TL;DR

This paper introduces the Singular Manifold Method, a unified framework leveraging the Painlevé Property to derive Lax pairs, transformations, and tau-functions for nonlinear PDEs, demonstrated on four (1+1)-dimensional equations.

Contribution

It presents a comprehensive approach that unifies the derivation of transformations and solutions for nonlinear PDEs using the Singular Manifold Method.

Findings

Derived Lax pairs and transformations for studied equations

Unified framework simplifies analysis of nonlinear PDEs

Applied method successfully to four (1+1)-dimensional equations

Abstract

This paper is an attempt to present and discuss at some length the Singular Manifold Method. This Method is based upon the Painlev\'e Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura, B\"acklund or Darboux Transformations as well as $\tau$-functions, in a unified way. Besides to present the basics of the Method we exemplify this approach by applying it to four equations in $(1+1)$-dimensions. Two of them are related with the other two through Miura transformations that are also derived by using the Singular Manifold Method.