On Soliton Interactions for a Hierarchy of Generalized Heisenberg Ferromagnetic Models on SU(3)/S(U(1) $\times$ U(2)) Symmetric Space

TL;DR

This paper develops an integrable hierarchy of nonlinear equations related to a symmetric space, constructs multi-soliton solutions, and analyzes soliton interactions, revealing unique properties of solutions with even and odd dispersion laws.

Contribution

It introduces a new hierarchy of generalized Heisenberg ferromagnetic models on a symmetric space and constructs explicit multi-soliton solutions using the Zakharov-Shabat dressing method.

Findings

Constructed N-soliton solutions for the hierarchy.

Identified two types of solitons: quadruplet and doublet.

Discovered that solutions with even dispersion laws are not traveling waves.

Abstract

We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is Z2 \times Z2 reduced and is naturally associated with the symmetric space SU(3)/S(U(1) \times U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the N-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one-soliton solutions of NLEEs with even and odd dispersion laws have different properties. In particular, the one-soliton solutions for NLEEs with even dispersion laws are not traveling waves; their velocities and their amplitudes are time dependent. Calculating the asymptotics of the N-soliton solutions for t \rightarrow \pm \infty we analyze the interactions of quadruplet solitons.