Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross-Pitaevskii Equation

TL;DR

This paper rigorously proves the existence of nonlinear Bloch waves bifurcating from the spectrum in the Gross-Pitaevskii equation with periodic potential, validating formal asymptotics and exploring their relation to other solutions.

Contribution

It provides a rigorous mathematical justification for the bifurcation of nonlinear Bloch waves and estimates the approximation error, expanding understanding of solutions in the GP equation.

Findings

Existence of nonlinear Bloch waves is rigorously established.

Formal asymptotics are validated with error estimates.

Numerical bifurcation diagrams illustrate the theoretical results.

Abstract

We rigorously analyze the bifurcation of stationary so called nonlinear Bloch waves (NLBs) from the spectrum in the Gross-Pitaevskii (GP) equation with a periodic potential, in arbitrary space dimensions. These are solutions which can be expressed as finite sums of quasi-periodic functions, and which in a formal asymptotic expansion are obtained from solutions of the so called algebraic coupled mode equations. Here we justify this expansion by proving the existence of NLBs and estimating the error of the formal asymptotics. The analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In addition, we illustrate some relations of NLBs to other classes of solutions of the GP equation, in particular to so called out--of--gap solitons and truncated NLBs, and present some numerical experiments concerning the stability of these solutions.