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

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.