Generalized Landau–Lifshitz models on the interval

Abstract

We study the classical generalized gl n Landau-Lifshitz (L-L) model with special boundary conditions that preserve integrability. We explicitly derive the first non-trivial local integral of motion, which corresponds to the boundary Hamiltonian for the sl 2 L-L model. Novel expressions of the modified Lax pairs associated to the integrals of motion are also extracted. The relevant equations of motion with the corresponding boundary conditions are determined. Dynamical integrable boundary conditions are also examined within this spirit. Then the generalized isotropic and anisotropic gl n Landau-Lifshitz models are considered, and novel expressions of the boundary Hamiltonians and the relevant equations of motion and boundary conditions are derived.

A dual construction of the isotropic Landau–Lifshitz model

“…Similarly, in order to extract the boundary conditions from the V -matrices, the condition that the equations of motion agree at the boundary manifests as the condition that lim x→±L VB,b = V±,b . Performing either of these limits yields the same constraint on the boundary constants and the S σ at the boundary [14]:…”

“…This is done for both closed (periodic) boundary conditions in Subsection 2.2 and open (reflective) boundary conditions in Subsection 2.3. Subsection 2.3 recalls the results of [14], except using notation that will be consistent with the sections that follow. Finally, we repeat these same steps for the dual (equal-space) construction of the HM model in Section 3, with the dual Poisson structure constructed in Subsection 3.1, and the hierarchies of dual Hamiltonians (and the corresponding Lax pairs) for both closed and open boundary conditions are constructed in Subsections 3.2 and 3.4, respectively.…”

“…The HM model has the same underlying r-matrix as the NLS model, namely the Yangian r-matrix (2.2), so hence it arises as a natural next step in the development of this dual approach. Because this equal-space picture follows in parallel to the usual method for building conserved quantities and higher systems (see [9]), we also introduce reflective time-like boundary conditions [10] to the HM model by following an equivalent procedure to the development of reflective space-like boundary conditions, [12,13], which have been applied to the isotropic Landau-Lifshitz equation in [14].…”