Killing fields and Conservation Laws for rank-1 Toda field equations

TL;DR

This paper explores the relationship between Killing fields and conservation laws in integrable systems, specifically for rank-1 Toda equations, providing a comprehensive method to generate conservation laws and analyze their properties.

Contribution

It establishes a connection between Killing fields and conservation laws, introduces a notion of finite-type for integral manifolds, and shows that all characteristic cohomology classes have translation-invariant representatives for rank-one Toda equations.

Findings

Generated complete set of conservation laws for the Tzitzeica equation.

Defined a generalized finite-type notion for integral manifolds.

Proved all characteristic cohomology classes have translation-invariant representatives.

Abstract

We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzeica equation, completing the work in Fox and Goertsches (2011). We define a notion of finite-type for integral manifolds of exterior differential systems directly in terms of conservation laws that generalizes the definition of Pinkall-Sterling (1989). The definition applies to any exterior differential system that has infinitely many conservation laws possessing a normal form. Finally, we show that, for the rank-one Toda field equations, every characteristic cohomology class has a translation invariant representative as an undifferentiated conservation law. Therefore the characteristic cohomology defines de Rham cohomology classes on doubly periodic solutions.