Generalized Landau-Lifshitz models on the interval

TL;DR

This paper investigates generalized Landau-Lifshitz models with boundary conditions that maintain integrability, deriving new boundary Hamiltonians, equations of motion, and Lax pairs for both isotropic and anisotropic cases.

Contribution

It provides explicit derivations of boundary Hamiltonians, integrals of motion, and Lax pairs for generalized Landau-Lifshitz models with special boundary conditions.

Findings

Explicit boundary Hamiltonians for generalized models

New expressions for Lax pairs and equations of motion

Analysis of boundary conditions preserving integrability

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.