Diffraction managed solitons: asymptotic validity and excitation thresholds

Abstract

Solitons in discrete optical systems have been a topic of keen interest over the past decade. In a diffraction managed waveguide array, where the waveguide's diffraction profile is varied periodically in the direction of propagation, the slow evolution of the electric field envelope is modelled by a nonlocal discrete equation that exhibits soliton solutions numerically. In this paper, we verify the validity of the asymptotic approximation, demonstrating that solutions of the averaged equation are close in the l2 sense to those of the original model for long time scales. Moreover, we prove that the averaged equation has stable ground state solutions, complementing previous experimental and numerical observations. Finally, we show that for a generalization of the averaged equation, ground states of an associated Hamiltonian exist if and only if their l2 norm exceeds a certain excitation threshold, which depends on the system's dimension and degree of nonlinearity.