Generalized negative flows in hierarchies of integrable evolution equations

TL;DR

This paper introduces a one-parameter generalization of negative flows in integrable hierarchies, expanding the class of nonlinear wave equations with preserved integrability properties like symmetries, conservation laws, and bi-Hamiltonian structure.

Contribution

It presents a novel one-parameter generalization of negative flows in integrable hierarchies, broadening the scope of integrable nonlinear wave equations with established properties.

Findings

Established integrability properties of generalized negative flows.

Explicit formulation for several classical integrable equations.

Extended the hierarchy of negative flows to a wider class of equations.

Abstract

A one-parameter generalization of the hierarchy of negative flows is introduced for integrable hierarchies of evolution equations, which yields a wider (new) class of non-evolutionary integrable nonlinear wave equations. As main results, several integrability properties of these generalized negative flow equation are established, including their symmetry structure, conservation laws, and bi-Hamiltonian formulation. (The results also apply to the hierarchy of ordinary negative flows). The first generalized negative flow equation is worked out explicitly for each of the following integrable equations: Burgers, Korteweg-de Vries, modified Korteweg-de Vries, Sawada-Kotera, Kaup-Kupershmidt, Kupershmidt.