Generalized Ablowitz-Ladik equation with a dual Lagrangian structure

Abstract

This article presents a generalized Ablowitz-Ladik (GAL) equation that can be obtained by means of the least action principle in two different ways. This GAL equation may describe the evolution of a discrete system, in which case it is derived from a Langrangian L which obeys a standard Lagrange equation. But this GAL equation may also describe a continuous system with nonlocal interactions, and in this case it is obtained from a different Lagrangian density L which obeys completely different Euler-Lagrange equations. The Lagrangian density L permits us to obtain variational solutions and Noether's conservation laws. Anderson's variational method predicts the existence of breathers and soliton-like solutions, and direct numerical solutions confirm the existence of these waves. The relationship between L and L defines a completely new type of “Lagrangian equivalence”, as one of them (L) describes the evolution of a discrete system, while the other (L) governs the behavior of a continuous nonlocal system.