Motivated by recent experimental studies of matter waves and optical beams in double-well potentials, we study the corresponding solutions of the nonlinear Schrödinger equation. Using a Galerkin-type approach, we obtain a detailed handle on the nonlinear solution branches of the problem, starting from the corresponding linear ones, and we predict the relevant bifurcations for both attractive and repulsive nonlinearities. The dynamics of the ensuing unstable solutions is also examined. The results illustrate the differences that arise between the steady states and the bifurcations emerging in symmetric and asymmetric double wells.

The dynamics of dark matter-wave solitons in elongated atomic condensates are discussed at finite temperatures. Simulations with the stochastic Gross-Pitaevskii equation reveal a noticeable, experimentally observable spread in individual soliton trajectories, attributed to inherent fluctuations in both phase and density of the underlying medium. Averaging over a number of such trajectories (as done in experiments) washes out such background fluctuations, revealing a well-defined temperature-dependent temporal growth in the oscillation amplitude. The average soliton dynamics is well captured by the simpler dissipative Gross-Pitaevskii equation, both numerically and via an analytically derived equation for the soliton center based on perturbation theory for dark solitons.

We examine an example of spontaneous symmetry breaking in a double-well waveguide with a symmetric potential. The ground state of the system beyond a critical power becomes asymmetric. The effect is illustrated numerically, and quantitatively analyzed via a Galerkin truncation that clearly shows the bifurcation from a symmetric to an asymmetric steady state. This phenomenon is also demonstrated experimentally when a probe beam is launched appropriately into an optically induced photonic lattice in a photorefractive material.

We consider a class of nonlinear Schrödinger / Gross-Pitaveskii (NLS-GP) equations, i.e. NLS with a linear potential. We obtain conditions for a symmetry breaking bifurcation in a symmetric family of states as N , the squared L 2 norm (particle number, optical power), is increased. In the special case where the linear potential is a doublewell with well separation L, we estimate N cr (L), the symmetry breaking threshold. Along the "lowest energy" symmetric branch, there is an exchange of stability from the symmetric to asymmetric branch as N is increased beyond N cr .

No abstract