Differential-algebraic approach to constructing representations of commuting differentiations in functional spaces and its application to nonlinear integrable dynamical systems

TL;DR

This paper develops a differential-algebraic method to represent commuting differentiations in functional spaces, applying it to construct Lax representations for integrable systems like Burgers and KdV equations, revealing their bi-Hamiltonian structures.

Contribution

It introduces a novel differential-algebraic framework for representing commuting differentiations and constructs Lax and bi-Hamiltonian hierarchies for nonlinear integrable systems.

Findings

Constructed Lax representations for Burgers and KdV type systems.

Analyzed a differential ideal with a conserved quantity.

Established an infinite bi-Hamiltonian hierarchy.

Abstract

There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only one conserved quantity is analyzed in detail, the corresponding Lax type representations of differentiations are constructed for an infinite hierarchy of nonlinear dynamical systems of the Burgers and Korteweg-de Vries type. A related infinite bi-Hamiltonian hierarchy of Lax type dynamical systems is constructed.