Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lipeng Duan, Suting Wei, Jun Yang

Abstract

We concern standing wave solutions with frequency $λ$ 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 $\{λ_j\}$ of the frequency $λ$ with $-λ_j\rightarrow +\infty$ 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].

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

Abstract

We concern standing wave solutions with frequency 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 of the frequency with 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].
Paper Structure (40 sections, 18 theorems, 686 equations)

This paper contains 40 sections, 18 theorems, 686 equations.

Table of Contents

  1. Introduction
  2. The problem with a constraint
  3. The singular perturbation problem
  4. Main results
  5. Necessary conditions for solutions with concentration
  6. Necessary conditions for concentration phenomena to the singular perturbation problem
  7. Necessary conditions for solutions with concentration to problem \ref{['eq2']}-\ref{['constraint1']}
  8. Existence of solutions with concentration phenomena
  9. Outline of the strategy and further comments
  10. Some preliminaries and preparing works
  11. The local forms of the equations in Fermi coordinates
  12. Setting up near the curves
  13. The block solution $U$ and conventions
  14. Proof of Theorem \ref{['necessary condition theorem']}
  15. The identities of Pohozaev type
  16. ...and 25 more sections

Key Result

Theorem 2.2

Suppose that: problem eq2-constraint1 has a solution $u_\epsilon$ which concentrates at the smooth simple closed curve $\Gamma_\epsilon$, i.e. the solution $u_\epsilon$ enjoys the decomposition u-decomposition plus the requirements in mathbfu-smallperturbation1. If lambda a to bf a and the assumptio (2). For the $\bf a$ given in lambda a to bf a, there holds (3). The following relation holds:

Theorems & Definitions (33)

  • Remark 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Lemma 4.1
  • ...and 23 more