Asymptotic soliton train solutions of the defocusing nonlinear Schrödinger equation

Abstract

Asymptotic behavior of initially ''large and smooth'' pulses is investigated at two typical stages of their evolution governed by the defocusing nonlinear Schrödinger equation. At first, wave breaking phenomenon is studied in the limit of small dispersion. A solution of the Whitham modulational equations is found for the case of dissipationless shock wave arising after the wave breaking point. Then, asymptotic soliton trains arising eventually from a large and smooth initial pulse are studied by means of a semiclassical method. The parameter varying along the soliton train is calculated from the generalized Bohr-Sommerfeld quantization rule, so that the distribution of eigenvalues depends on two functions-intensity 0 (x) of the initial pulse and its initial chirp v 0 (x). The influence of the initial chirp on the asymptotic state is investigated. Excellent agreement of the numerical solution of the defocusing NLS equation with predictions of the asymptotic theory is found.

Formation of shock waves in a Bose-Einstein condensate

“…It is also worth to mention, that it is possible to find an approximate analytical solution of hydrodynamical equations (1) with the quantum pressure term. That solution also exhibits shock behavior, and its properties have been discussed in [13,14,15].…”

“…The result (5) is valid as long as a spatial scale of density perturbations is larger than the healing length ξ. Solutions of the form (5) were previously worked out in various physical systems, see e. g. [13,14,15]. Nonlinearity of (1) prevents usage of the superposition principle to built a solution, that contains both left and right-moving perturbations, out of (5).…”

“…[19]. Later on, the solution (6) in a shock wave front region becomes multi-valued, compare [13,14,15].…”

“…The propagation of steep density perturbations in a BEC has been experimentally investigated in [11]; the theory of shocks in Fermi (Tonks) gas has been proposed in [12]; an approximate theoretical description of shock waves in a BEC has been worked out in [13]. It is also worth to mention that shocks have been studied in physical systems governed by nonlinear Schrödinger equation [14,15]. These studies can be related to investigations of shock waves in a BEC due to formal similarity between the nonlinear Schrödinger equation and mean-field equations of a BEC.…”

Crossover dynamics of dispersive shocks in Bose-Einstein condensates characterized by two- and three-body interactions

“…In this paper we address this question with reference to the dynamics of dispersive shock waves (DSWs), which form in a repulsive BEC when the kinetic spreading regularizes the tendency driven by the nonlinearity to form multivalued wave fronts. At variance with earlier [3,4,16] and more recent studies on DSWs [17][18][19][20], which are all based on the standard GPE, we * www.primalight.org; andrea.fratalocchi@kaust.edu.sa investigate the case characterized by an attractive three-body nonlinear correction to the repulsive s-wave scattering. This regime is expected to be achievable by exploiting resonance tuning for bosons and turns out to be relevant also for superfluid fermions [21].…”