A Geometric Method to Investigate Prolongation Structures for Differential Systems With Applications to Integrable Systems

TL;DR

This paper introduces a geometric method for analyzing prolongation structures in differential systems, focusing on Lie algebra-based forms, and demonstrates how these structures can be explicitly constructed and reduced to simpler problems, revealing conservation laws.

Contribution

It presents a novel geometric approach to construct and analyze prolongation structures for differential systems with Lie algebra symmetries, including explicit reduction techniques.

Findings

Prolongation structures are explicitly constructed for systems with SL(2,R), O(3), and SU(3) symmetries.

The 3x3 systems can be reduced to three 2x2 problems, simplifying analysis.

Multiple conservation laws are derived at various stages of the development.

Abstract

A type of prolongation structure for several general systems is discussed. They are based on a set of one-forms in which the underlying structure group of the integrability condition corresponds to the Lie-algebra of SL (2,R), O(3), or SU(3). Each will be considered in turn and the latter two systems represent larger 3by3 cases. This geometric approach is applied to all three of these systems to obatin prolongation structures explicitly. In both 3by3 cases the prolongation structure is reduced to the situation of three smaller 2by2 problems. Many types of conservation laws can be obtained at different stages of the development, and at the end, a single result is developed to show how this can be done.