Spikes in two coupled nonlinear Schrödinger equations. (English) Zbl 1080.35143
Summary: Here we study the interaction and the configuration of spikes in a double condensate by analyzing least energy solutions of two coupled nonlinear Schrödinger equations which model Bose-Einstein condensates of two different hyperfine spin states. When the interspecies scattering length is positive and large enough, spikes of a double condensate repel each other and behave like two separate spikes. In contrast, spikes of a double condensate attract each other and behave like a single spike if the interspecies scattering length is negative and large enough. Our mathematical arguments can prove such physical phenomena. We first use Nehari’s manifold to construct least energy solutions, and then use some techniques of singular perturbation problems to derive the asymptotic behavior of least energy solutions. It is shown that the interaction term determines the locations of the two spikes and the asymptotic shape of least energy solutions.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 82D10 | Statistical mechanics of plasmas |
| 35B45 | A priori estimates in context of PDEs |
| 35J40 | Boundary value problems for higher-order elliptic equations |
Keywords:
Hartree-Fock theory; Fermi gas; Nehari’s manifold; least-energy solution; Bose-Einstein condensatesReferences:
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