Domain Walls in the Coupled Gross–Pitaevskii Equations

Abstract

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 minimizing 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.

Vortex patterns and sheets in segregated two component Bose–Einstein condensates

“…Proof. It follows from [6], Theorem 3.1, that there exists a minimizer for problem (2.6)-(2.7). Moreover, each minimizer satisfies that each component is monotone.…”

Monotonicity and rigidity of solutions to some elliptic systems with uniform limits

“…As byproduct of Theorem 1.1, we obtain the uniqueness (modulo translations) of the positive 1D heteroclinic connections between (0, 1) and (1, 0), without any additional assumption on (u, v). For minimal solutions (in a suitable sense), this result was conjectured in [2,Section 5], and proved in [1] as a consequence of [1, Theorem 1.3], where it is showed the uniqueness of positive 1D heteroclinic connections with one monotone component. Here we can remove the monotonicity assumption and extend the uniqueness in any dimension.…”

Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

“…This system arises in the study of domain walls and interface layers for two-components Bose-Einstein condensates [4]. Domain walls solutions satisfying asymptotic conditions (1.2) (u, v) → (1, 0) as x N → +∞, (u, v) → (0, 1) as x N → −∞, , in dimension N = 1 have been carefully studied in [2,4], where in particular it is shown the existence of such a solution for every Λ > 1 [4], and its uniqueness in the class of solutions with one monotone component [2]. In fact, uniqueness holds also without such assumption, and even in higher dimension [9]; precisely, in [9] it is shown that a solution to (1.1)-(1.2) (with the limits being uniform in x ′ ∈ R N −1 ) in R N with Λ > 1 is necessarily montone in both the components with respect to x N , and 1-dimensional.…”