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.


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