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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
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