Particle scattering and fusion for the Ablowitz–Ladik chain

Abstract

The Ablowitz–Ladik (AL) 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 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 AL 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.