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

• We study nonlinear states for the NLS-type equation with additional periodic potential.
• In some cases all the nonlinear states can be coded by bi-infinite sequences.
• Sufficient conditions for the coding to be possible are formulated.
• The approach is illustrated by the case of NLS with the cosine potential.

We study nonlinear states for the NLS-type equation with additional periodic potential U(x)U(x), also called the Gross–Pitaevskii equation, GPE  , in theory of Bose–Einstein Condensate, BEC. We prove that if the nonlinearity is defocusing (repulsive, in the BEC context) then under some conditions there exists a homeomorphism between the set of all 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 NN for some integer NN. These sequences can be viewed as codes of the nonlinear states. We present numerical arguments that for GPE with cosine potential these conditions hold in certain areas of the plane of the external parameters. This implies that for these values of parameters all the nonlinear states can be described in terms of the coding sequences.