Nonlinear Kr\"onig-Penney model

TL;DR

This paper analyzes a nonlinear Schrödinger equation with a periodic delta potential, deriving analytical solutions for Bloch states and exploring new periodic solutions relevant to Bose-Einstein condensates and optical fibers.

Contribution

It introduces analytical solutions for nonlinear Bloch states in a nonlinear Krönig-Penney model, including states with different periodicities from the potential.

Findings

Derived zero-current Bloch state solutions.

Identified new classes of solutions with different periodicities.

Compared chemical potential of nonlinear states with linear spectrum.

Abstract

We study the nonlinear Schr\"odinger equation with a periodic delta-function potential. This realizes a nonlinear Kr\"onig-Penney model, with physical applications in the context of trapped Bose-Einstein condensate alkaly gases and in the transmission of signals in optical fibers. We find analytical solutions of zero-current Bloch states. Such wave-functions have the same periodicity of the potential, and, in the linear limit, reduce to the Bloch functions of the Kr\"onig-Penney model. We also find new classes of solutions having a periodicity different from that of the external potential. We calculate the chemical potential of such states and compare it with the linear excitation spectrum.