Geometric Approaches for Generating Prolongations for Nonlinear Partial Differential Equations

TL;DR

This paper introduces geometric methods using exterior differential forms and Lie algebra structures to generate prolongations for nonlinear partial differential equations, providing a general framework for integrability analysis.

Contribution

It develops a general geometric framework for prolongation structures of nonlinear PDEs using exterior differential forms and Lie algebra, without relying on specific equations.

Findings

Formulation of prolongation structures via exterior differential forms.

Derivation of integrability conditions involving SU(2) Lie algebra.

Discussion of Wahlquist-Estabrook prolongation and applications.

Abstract

The prolongation structure of a two-by-two problem is formulated very generally in terms of exterior differential forms on a standard representation of Pauli matrices. The differential system is general without making reference to any specific equation. An integrability condition is provided which gives by construction the equation to be investigated and whose components involve the structure constants of an SU(2) Lie algebra. Along side this, a related, different kind of prolongation, a type of Wahlquist-Estabrook prolongation, over a closed differential ideal is discussed and some applications are given.