Exact solutions for periodic and solitary matter waves in nonlinear lattices

TL;DR

This paper derives exact periodic and soliton solutions for the one-dimensional Gross-Pitaevskii equation with nonlinear lattices, analyzing their stability and conditions for persistence in Bose-Einstein condensates.

Contribution

It introduces three classes of exact solutions using Jacobi elliptic functions for the GPE with nonlinear lattices, including stability analysis and parameter characterization.

Findings

Stable cn-type solutions can exist with or without linear potential.

dn and sn solutions require linear potential for stability.

Solutions reduce to solitons as the period approaches infinity.

Abstract

We produce three vast classes of exact periodic and soliton solutions to the one-dimensional Gross-Pitaevskii equation (GPE) with the pseudopotential in the form of a nonlinear lattice (NL), induced by a spatially periodic modulation of the local nonlinearity. It is well known that NLs in Bose-Einstein condensates (BECs) may be created by means of the Feshbach-resonance technique. The model may also include linear potentials with the same periodicity. The NL modulation function, the linear potential (if any), and the corresponding exact solutions are expressed in terms of the Jacobi's elliptic functions of three types, cn, dn, and sn, which give rise to the three different classes of the solutions. The potentials and associated solutions are parameterized by two free constants and an additional sign parameter in the absence of the linear potential. In the presence of the latter, the solution families feature two additional free parameters. The families include both sign-constant and sign-changing NLs. Density maxima of the solutions may coincide with either minima or maxima of the periodic pseudopotential. The solutions reduce to solitons in the limit of the infinite period. The stability of the solutions is tested via direct simulations of the GPE. As a result, stability regions are identified for the periodic solutions and solitons. The periodic patterns of the cn type, and the respective limit-form solutions in the form of bright solitons, may be stable both in the absence and presence of the linear potential. The stability of the two other solution classes, of the dn and sn types, is only possible with the linear potential.