Particle Scattering and Fusion for the Ablowitz-Ladik Chain

Abstract

The Ablowitz-Ladik chain is an integrable discretized version of the nonlinear Schrödinger equation. We report on a novel underlying Hamiltonian particle system with properties similar to the ones known for the classical Toda chain and Calogero fluid with $1/{\sinh }^2$ pair interaction. Boundary conditions are imposed such that, both in the distant past and future, particles have a constant velocity. We establish the many-particle scattering for the Ablowitz-Ladik chain and obtain properties known for generic integrable many-body systems. For a specific choice of the chain, real initial data remain real in the course of time. Then, asymptotically, particles move in pairs with a velocity-dependent size and scattering shifts are governed by the fusion rule.

32 pages, 5 figures, 2 appendices

References in corpus (6)

• Introduction to the thermodynamic Bethe ansatz
• Generalized Hydrodynamics of the attractive Non-Linear Schroedinger Equation
• Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale
• Large Deviations for Ablowitz-Ladik lattice, and the Schur flow
• Discrete integrable systems and random Lax matrices
• Generalized Hydrodynamics for the Volterra lattice: Ballistic and nonballistic behavior of correlation functions

Cited by in corpus (2)

• Hydrodynamics of spin transport in the Haldane-Shastry chain
• Lax random matrices from Calogero systems