Solutions with one dimensional concentration for a two dimensional Gross-Pitaevskii model with general potential

TL;DR

This paper constructs solutions to a 2D Gross-Pitaevskii equation with a trap potential, showing concentration along curves and addressing a conjecture about solutions concentrating at submanifolds.

Contribution

It provides necessary conditions for solution concentration along curves and constructs solutions with frequency sequences diverging to infinity, partially answering a prior conjecture.

Findings

Solutions concentrate along smooth curves under certain conditions.

Constructed solutions with diverging frequency sequences.

Addresses a conjecture on submanifold concentration.

Abstract

We concern standing wave solutions with frequency $\lambda$ to a two dimensional Gross-Pitaevskii equation with a trap potential under the unit mass constraint, which is used to describe Bose-Einstein condensates with attractive interaction. First, we investigate the necessary conditions for existence of the solutions with concentration phenomena directed along closed smooth curves. Next, not only imposing stationary and non-degeneracy conditions on the curves with respect to an auxiliary weighted length involving the trap potential, but also adding some other technical assumptions, we select a sequence $\{\lambda_j\}$ of the frequency $\lambda$ with $-\lambda_j\rightarrow +\infty$ and construct solutions with concentration directed along the curves. Our result partially answers the conjecture raised in [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Comm. Math. Phys. 2003] about necessary condition for solution concentrating at submanifolds. The solutions constructed in this paper are concentrating on curves whose length are non-uniformly bounded, and hence the situation is quite different from that in [M. del Pino, M. Kowalczyk, J. Wei, Comm. Pure Appl. Math. 2007].