Two-dimensional paradigm for symmetry breaking: the nonlinear Schrödinger equation with a four-well potential.
We study the existence and stability of localized modes in the two-dimensional (2D) nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation with a symmetric four-well potential. Using the corresponding four-mode approximation, we trace the parametric evolution of the trapped stationary modes, starting from the linear limit, and thus derive a complete bifurcation diagram for families of the stationary modes. This provides the picture of spontaneous symmetry breaking in the fundamental 2D setting. In a broad parameter region, the predictions based on the four-mode decomposition are found to be in good agreement with full numerical solutions of the NLS/GP equation. Stability properties of the stationary states coincide with those suggested by the corresponding discrete model in the large-amplitude limit. The dynamics of unstable modes is explored by means of direct simulations. Finally, in addition to the full analysis for the case of the self-attractive nonlinearity, the bifurcation diagram for the case of self-repulsion is briefly considered too.
Wang
C
C
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA.
Theocharis
G
G
Kevrekidis
P G
PG
Whitaker
N
N
Law
K J H
KJ
Frantzeskakis
D J
DJ
Malomed
B A
BA
D000465:Algorithms; D003198:Computer Simulation; D008027:Light; D008962:Models, Theoretical; D017711:Nonlinear Dynamics; D017785:Photons; D011789:Quantum Theory