Notes on Exact Multi-Soliton Solutions of Noncommutative Integrable Hierarchies

TL;DR

This paper investigates exact multi-soliton solutions in noncommutative integrable hierarchies, demonstrating that their asymptotic behavior and phase shifts mirror those in commutative cases, and introduces a noncommutative Gelfand-Dickey hierarchy with explicit solutions.

Contribution

It provides a detailed analysis of multi-soliton solutions in noncommutative spaces and introduces a new noncommutative Gelfand-Dickey hierarchy with exact solutions.

Findings

Asymptotic soliton configurations are identical to commutative cases.

Solitary waves preserve shape and velocity during scattering.

Exact solutions for noncommutative Gelfand-Dickey hierarchy are constructed.

Abstract

We study exact multi-soliton solutions of integrable hierarchies on noncommutative space-times which are represented in terms of quasi-determinants of Wronski matrices by Etingof, Gelfand and Retakh. We analyze the asymptotic behavior of the multi-soliton solutions and found that the asymptotic configurations in soliton scattering process can be all the same as commutative ones, that is, the configuration of N-soliton solution has N isolated localized energy densities and the each solitary wave-packet preserves its shape and velocity in the scattering process. The phase shifts are also the same as commutative ones. Furthermore noncommutative toroidal Gelfand-Dickey hierarchy is introduced and the exact multi-soliton solutions are given.