S. Alama, L. Bronsard, A. Contreras, and D.E. Pelinovsky
Domain walls in the coupled Gross-Pitaevskii equations
Archives for Rational Mechanics and its Applications 215, 579-610 (2015)
A thorough study of domain wall solutions in coupled Gross-Pitaevskii equations on the real line
is carried out including existence of these solutions; their spectral and nonlinear stability;
their persistence and stability under a small localized potential. The proof of existence
is variational and is presented in a general framework: we show that the domain wall solutions
are energy minimizers within a class of vector-valued functions with nontrivial conditions at infinity.
The admissible energy functionals include those corresponding to coupled
Gross-Pitaevskii equations, arising in modeling of Bose-Einstein condensates.
The results on spectral and nonlinear stability follow from properties
of the linearized operator about the domain wall. The methods apply to many systems of interest
and integrability is not germane to our analysis. Finally, sufficient conditions for
persistence and stability of domain wall solutions are obtained to show that stable pinning
occurs near maxima of the potential, thus giving rigorous justification to earlier results
in the physics literature.
coupled Gross--Pitaevskii equations, domain walls, existence and stability, variational method, Lyapunov-Schmidt reductions.