Freezing of Nonlinear Bloch Oscillations in the Generalized Discrete Nonlinear Schrodinger Equation

The dynamics in a nonlinear Schrodinger chain in an 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 phenomenon.Comment: LaTeX, 7 pages, 6 figures. Improved version to appear in Phys. Rev.

Whitham method for Benjamin-Ono-Burgers equation and dispersive shocks in internal waves in deep fluid

The Whitham modulation equations for the parameters of a periodic solution are derived using the generalized Lagrangian approach for the case of damped Benjamin-Ono equation. The structure of the dispersive shock in internal wave in deep water is considered by this method.Comment: 8 pages, 4 figure

On dissipationless shock waves in a discrete nonlinear Schr\"odinger equation

It is shown that the generalized discrete nonlinear Schr\"odinger equation can be reduced in a small amplitude approximation to the KdV, mKdV, KdV(2) or the fifth-order KdV equations, depending on values of the parameters. In dispersionless limit these equations lead to wave breaking phenomenon for general enough initial conditions, and, after taking into account small dispersion effects, result in formation of dissipationless shock waves. The Whitham theory of modulations of nonlinear waves is used for analytical description of such waves.Comment: 15 pages, 9 figure

Global existence of small-norm solutions in the reduced Ostrovsky equation

We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitz\'eica equation and prove global existence of small-norm solutions in Sobolev space $H^3(R)$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.Comment: 11 pages; 1 figur

Soliton annihilation into a polychromatic dispersive wave

International audienceWe investigate the propagation of a soliton in an axially-varying optical fiber with a progressive change from anomalous to normal dispersion regimes. Spectral and temporal measurements provide evidence for a complete annihilation of the soliton, which explodes into a polychromatic dispersive wave. This interpretation is confirmed by numerical solution of the generalized nonlinear Schrödinger equation