Interaction of modulated pulses in the nonlinear Schroedinger equation with periodic potential (original) (raw)
Nonlinear dynamics of semiclassical coherent states in periodic potentials
Journal of Physics A: Mathematical and Theoretical, 2012
We consider nonlinear Schrödinger equations with either local or nonlocal nonlinearities. In addition, we include periodic potentials as used, for example, in matter wave experiments in optical lattices. By considering the corresponding semiclassical scaling regime, we construct asymptotic solutions, which are concentrated both in space and in frequency around the effective semiclassical phase-space flow induced by Bloch's spectral problem. The dynamics of these generalized coherent states is governed by a nonlinear Schrödinger model with effective mass. In the case of nonlocal nonlinearities we establish a novel averaging type result in the critical case.
Journal of Mathematical Analysis and Applications, 2017
We consider wavepackets composed of two modulated carrier Bloch waves with opposite group velocities in the one dimensional periodic Nonlinear Schrödinger/Gross-Pitaevskii equation. These can be approximated by first order coupled mode equations (CMEs) for the two slowly varying envelopes. Under a suitably selected periodic perturbation of the periodic structure the CMEs possess a spectral gap of the corresponding spatial operator and allow families of exponentially localized solitary waves parametrized by velocity. This leads to a family of approximate solitary waves in the periodic nonlinear Schrödinger equation. Besides a formal derivation of the CMEs a rigorous justification of the approximation and an error estimate in the supremum norm are provided. Several numerical tests corroborate the analysis.
Stability and decay of Bloch oscillations in the presence of time-dependent nonlinearity
Physical Review A, 2011
We consider Bloch oscillations of Bose-Einstein condensates in the presence of a time-modulated s-wave scattering length. Generically, the interaction leads to dephasing and decay of the wave packet. Based on a cyclic-time argument, we find-in addition to the linear Bloch oscillation and a rigid soliton solution-an infinite family of modulations that lead to a periodic time evolution of the wave packet. In order to quantitatively describe the dynamics of Bloch oscillations in the presence of time-modulated interactions, we employ two complementary methods: collective coordinates and the linear stability analysis of an extended wave packet. We provide instructive examples and address the question of robustness against external perturbations.
Interaction of modulated pulses in scalar multidimensional nonlinear lattices
Applicable Analysis, 2010
We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitrary-range, nonlinear interaction potentials and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high order of the scaling parameter, thereby deducing the resonance and non-resonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets of up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.
Freezing of nonlinear Bloch oscillations in the generalized discrete nonlinear Schrödinger equation
Physical Review E, 2004
The dynamics in a nonlinear Schrödinger chain in a homogeneous electric field is studied. We show that discrete translational invariant integrability-breaking terms can freeze the Bloch nonlinear oscillations and introduce new faster frequencies in their dynamics. These phenomena are studied by direct numerical integration and through an adiabatic approximation. The adiabatic approximation allows a description in terms of an effective potential that greatly clarifies the phenomena.
Modulational Instability of Double-Periodic Waves in the Nonlinear Schrödinger Equation
2020
It is shown how to compute the modulational instability rates for the doubleperiodic solutions to the cubic NLS (nonlinear Schrödinger) equation by using the Lax linear equations. The wave function modulus of the double-periodic solutions is periodic both in space and time coordinates; such solutions generalize the standing waves which have the time-independent and space-periodic wave function modulus. Similar to other waves in the NLS equation, the double-periodic solutions are modulationally unstable and this instability is related to the bands of the Lax spectrum outside the imaginary axis. A simple numerical method is used to compute the modulational instability bands and to compare the instability rates of the double-periodic solutions with those of the standing periodic waves.
Nonlinear Schrödinger equation with spatiotemporal perturbations
Physical Review E, 2010
We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation ͑NLSE͒ with the following perturbations: nonparametric spatiotemporal driving of the form f͑x , t͒ = a exp͓iK͑t͒x͔, damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a collective-coordinate-theory which yields a set of ordinary differential equations ͑ODEs͒ for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force f͑x͒. The soliton position exhibits oscillations around a mean trajectory with constant velocity. This means that the soliton performs, on the average, a unidirectional motion although the spatial average of the force vanishes. The amplitude of the oscillations is much smaller than the period of f͑x͒. In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum P͑t͒ and the soliton velocity V͑t͒: This is a parameter representation of a curve P͑V͒ which is visited by the soliton while time evolves. Our conjecture is that the soliton becomes unstable, if this curve has a branch with negative slope. This conjecture is fully confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the soliton lives longer, the shorter the branch with negative slope is.