Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential

Abstract

We study the existence and stability of localized states in the two-dimensional (2D) nonlinear Schrödinger (NLS)/Gross-Pitaevskii equation with a symmetric four-well potential. Using a fourmode approximation, we are able to trace the parametric evolution of the trapped stationary modes, starting from the corresponding linear limits, and thus derive the complete bifurcation diagram for the families of these stationary modes. The predictions based on the four-mode decomposition are found to be in good agreement with the numerical results obtained from the NLS equation. Actually, the stability properties coincide with those suggested by the corresponding discrete model in the large-amplitude limit. The dynamics of the unstable modes is explored by means of direct simulations. Finally, while we present the full analysis for the case of the focusing nonlinearity, the bifurcation diagram for the defocusing case is briefly considered too.

Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

“…As μ → μ 0 , these two new solution branches merge with the smooth branch. Examples of pitchfork bifurcations reported so far are all symmetry-breaking bifurcations [11,[13][14][15][16][17][18][19], where a smooth branch of symmetric or antisymmetric solitary waves exists on both sides of the bifurcation point, but two new branches of asymmetric solutions appear on only one side of the bifurcation point. A transcritical bifurcation is where there are two smooth branches of solitary waves which exist on both sides of the bifurcation point μ 0 , and these solutions on both branches approach each other as μ → μ 0 .…”

“…Indeed, various solitary wave bifurcations in miscellaneous nonlinear wave models have been reported. Examples include saddle-node bifurcations (also called fold bifurcations) [2][3][4][5][6][7][8][9][10][11], pitchfork bifurcations (sometimes called symmetry-breaking bifurcations) [11][12][13][14][15][16][17][18][19], transcritical bifurcations [7], and so on. Most of these reports on bifurcations are numerical.…”

Existence, Stability and Dynamics of Nonlinear Modes in a 2D PartiallyPT Symmetric Potential

“…It is from these modes that we will seek bifurcations of nonlinear solutions in what follows. It is worthwhile to mention here, in comparison, e.g., with the real four-well potential of [23] that the latter possessed four localized modes, with the fourth antisymmetric, quadrupolar one being absent from the point spectrum in the case of interest herein. …”

Energy splitting in a finite periodic multiple-well potential

“…[12] (see, also, Ref. [20]), we also consider a two-dimensional potential in which the quadratic minima are located on the vertices of a regular N -sided polygon with rotational symmetry of order N . Through the N -level approximation, it is shown that, the lowest band of the two-dimensional model closely resembles the tight-binding energy bands of the fully periodic system [7,8], in that most of the eigenvalues are degenerate in the large-N limit with the eigenfunctions satisfying the Bloch condition under the discrete rotations.…”