Coding of nonlinear states for the Gross-Pitaevskii equation with periodic potential

TL;DR

This paper establishes a method to encode nonlinear states of the Gross-Pitaevskii equation with periodic potential using symbolic sequences, providing a new way to classify solutions in Bose-Einstein Condensate models.

Contribution

It introduces a homeomorphism between nonlinear states and symbolic sequences for the GPE with periodic potential, under certain verifiable conditions.

Findings

Numerical verification of hypotheses for cosine potential

Identification of parameter regions allowing coding of states

Establishment of a topological classification framework

Abstract

We study nonlinear states for NLS-type equation with additional periodic potential U(x) (called the Gross-Pitaevskii equation, GPE, in theory of Bose-Einstein Condensate, (BEC)). We prove that if the nonlinearity is defocusing (repulsive, in BEC context) then under certain conditions there exists a homeomorphism between the set of nonlinear states for GPE (i.e. real bounded solutions of some nonlinear ODE) and the set of bi-infinite sequences of numbers from 1 to N for some integer N. These sequences can be viewed as codes of the nonlinear states. Sufficient conditions for the homeomorphism to exist are given in the form of three hypotheses. For a given U(x), the verification of the hypotheses should be done numerically. We report on numerical results for the case of GPE with cosine potential and describe regions in the plane of parameters where this coding is possible.