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Archive for Rational Mechanics and Analysis

, Volume 190, Issue 1, pp 83–106 | Cite as

Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations

  • Juncheng Wei
  • Tobias WethEmail author
Article

Abstract

We consider the nonlinear elliptic system
$$\left \{ \begin{aligned} -&\Delta u +u - u^3 -\beta v^2u = 0\quad \rm{in}\, \mathbb B,\\ -&\Delta v +v - v^3 -\beta u^2v = 0\quad \rm{in}\, \mathbb B,\\ &u,v > 0 \quad \rm{in}\, \mathbb B,\quad u=v=0 \quad \rm{on}\, \partial \mathbb B, \end{aligned} \right.$$
where \(N\leqq 3\) and \(\mathbb B \subset \mathbb {R}^N\) is the unit ball. We show that, for every \(\beta \leqq -1\) and \(k \in \mathbb N\), the above problem admits a radially symmetric solution (u β , v β ) such that u β v β changes sign precisely k times in the radial variable. Furthermore, as \(\beta \to -\infty\), after passing to a subsequence, u β w + and v β w uniformly in \(\mathbb B\), where w = w +w has precisely k nodal domains and is a radially symmetric solution of the scalar equation Δww + w 3 = 0 in \(\mathbb B\), w = 0 on \(\partial \mathbb B\). Within a Hartree–Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains for Bose–Einstein double condensates with strong repulsion.

Keywords

Solitary Wave Einstein Condensate Radial Variable Radial Solution Nodal Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsChinese University of Hong KongShatinHong Kong
  2. 2.Mathematische InstitutUniversitat GiessenGiessenGermany

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