Ground-state Properties of Small-Size Nonlinear Dynamical Lattices

TL;DR

This paper studies the ground state properties of small nonlinear lattices using the discrete nonlinear Schrödinger equation, revealing unique features like coexistence of solutions and a novel bifurcation type absent in large lattices.

Contribution

It uncovers nontrivial ground state behaviors and a new bifurcation phenomenon specific to small nonlinear lattices, expanding understanding beyond large lattice limits.

Findings

Coexistence of single-pulse and uniform solutions in a finite coupling range.

Existence of a critical threshold beyond which the single-pulse mode disappears.

Identification of a novel 'double transcritical' bifurcation associated with ground state instability.

Abstract

We investigate the ground state of a system of interacting particles in small nonlinear lattices with M > 2 sites, using as a prototypical example the discrete nonlinear Schroedinger equation that has been recently used extensively in the contexts of nonlinear optics of waveguide arrays, and Bose-Einstein condensates in optical lattices. We find that, in the presence of attractive interactions, the dynamical scenario relevant to the ground state and the lowest-energy modes of such few-site nonlinear lattices reveals a variety of nontrivial features that are absent in the large/infinite lattice limits: the single-pulse solution and the uniform solution are found to coexist in a finite range of the lattice intersite coupling where, depending on the latter, one of them represents the ground state; in addition, the single-pulse mode does not even exist beyond a critical parametric threshold. Finally, the onset of the ground state (modulational) instability appears to be intimately connected with a non-standard (``double transcritical'') type of bifurcation that, to the best of our knowledge, has not been reported previously in other physical systems.