The nonlinear Schroedinger equation for the delta-comb potential: quasi-classical chaos and bifurcations of periodic stationary solutions
arXiv:cond-mat/0506645 · doi:10.1142/S1402925109000145
Abstract
The nonlinear Schroedinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schroedinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation of new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.
Enhanced and revised version, to appear in J. Nonlin. Math. Phys
References in corpus (23)
- Many-Body Physics with Ultracold Gases
- Direct Observation of Tunneling and Nonlinear Self-Trapping in a single Bosonic Josephson Junction
- Bright gap solitons of atoms with repulsive interaction
- Nonlinear Waves in Bose-Einstein Condensates: Physical Relevance and Mathematical Techniques
- Observation of dynamical instability for a Bose-Einstein condensate in a moving 1D optical lattice
- Superfluidity of Bose-Einstein Condensate in An Optical Lattice: Landau-Zener Tunneling and Dynamical Instability
- Adiabatic Theory of Nonlinear Evolution of Quantum States
- Nonlinear Resonant Transport of Bose Einstein Condensates
- Beyond mean-field dynamics of small Bose-Hubbard systems based on the number-conserving phase space approach
- Superfluidity versus Anderson localization in a dilute Bose gas
- Towards a Landau-Zener formula for an interacting Bose-Einstein condensate
- Nonlinear Band Structure in Bose Einstein Condensates: The Nonlinear Schrödinger Equation with a Kronig-Penney Potential
- Bound and resonance states of the nonlinear Schroedinger equation in simple model systems
- Nonlinear modes for the Gross-Pitaevskii equation -- demonstrative computation approach
- Mean-field dynamics of a two-mode Bose-Einstein condensate subject to noise and dissipation
- Bloch oscillations of Bose-Einstein condensates: Breakdown and revival
- Discovery of exceptional points in the Bose-Einstein condensation of gases with attractive 1/r-interaction
- Period doubling, two-color lattices, and the growth of swallowtails in Bose-Einstein condensates
- An analytical study of resonant transport of Bose-Einstein condensates
- Stability of Bose-Einstein condensates in a Kronig-Penney potential
- Nonlinear Krönig-Penney model
- Bose-Einstein condensates in accelerated double-periodic optical lattices: Coupling and Crossing of resonances
- Resonance solutions of the nonlinear Schrödinger equation in an open double-well potential
Cited by in corpus (12)
- Model of a PT symmetric Bose-Einstein condensate in a delta-functions double well
- Nonlinear Schrödinger equation for a PT symmetric delta-functions double well
- Coupling approach for the realization of a -symmetric potential for a Bose-Einstein condensate in a double well
- Spectral singularities in PT-symmetric Bose-Einstein condensates
- Quantum dynamics of Bose-Einstein condensates in tilted and driven bichromatic optical lattices
- Realizing PT-symmetric BEC subsystems in closed hermitian systems
- Coding of nonlinear states for the Gross-Pitaevskii equation with periodic potential
- Exceptional points in bichromatic Wannier-Stark systems
- Multiple-scale analysis for resonance reflection by a one-dimensional rectangular barrier in the Gross-Pitaevskii problem
- Multi-barrier resonant tunneling for the one-dimensional nonlinear Schrödinger Equation
- Transmission resonances in above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential
- Higher order transmission resonances in above-barrier reflection of ultra-cold atoms