On Soliton Interactions for the Hierarchy of a Generalised Heisenberg Ferromagnetic Model on ${\rm SU(3)/S(U(1)\times U(2))}$ Symmetric Space

Abstract

We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator $L$. The Lax representation is $\mathbb{Z}_2\times\mathbb{Z}_2$ reduced and can be naturally associated with the symmetric space ${\rm SU(3)/S(U(1)\times U(2))}$. 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 while their velocities and amplitudes are time dependent. Calculating the asymptotics of the $N$-soliton solutions for $t\to\pm\infty$ we analyze the interactions of quadruplet solitons.