## On soliton interactions for the hierarchy of a generalised Heisenberg ferromagnetic model on $$SU (3)/ S (U (1) \times U (2))$$ symmetric space.(English)Zbl 1259.82127

Summary: 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 $$SU (3)/ S (U (1) \times U (2))$$. The simplest nontrivial equation in the hierarchy is a generalization of the Heisenberg ferromagnetic model. We construct the $$N$$-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabar 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.

### MSC:

 82D40 Statistical mechanics of magnetic materials 35Q51 Soliton equations 17B80 Applications of Lie algebras and superalgebras to integrable systems 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures