When posed on a periodic domain in one space variable, linear dispersive evolution equations with integral polynomial dispersion relations exhibit strikingly different behaviors depending upon when the time is rational or irrational relative to the length of the interval: the Talbot phenomenon of dispersive quantization and fractalization. The purpose of this paper is to show that these phenome...

We consider the Schrödinger equation with no radial assumption on real hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we obtain strong wellposedness results for NLS. Specifically, for small intial data, we prove L 2 and H 1 global wellposedness for any subcritical nonlinearity (in contrast with the flat case...

We present a new algorithm, the Time Dependent Phase Space Filter (TDPSF) which is used to solve time dependent Nonlinear Schrodinger Equations (NLS). The algorithm consists of solving the NLS on a box with periodic boundary conditions (by any algorithm). Periodically in time we decompose the solution into a family of coherent states. Coherent states which are outgoing are deleted, while those ...

A method has been developed for deriving the approximate global optimum of a nonlinear objective function. First, the objective function is expanded into a linear equation for a moment vector, and the optimization problem is reduced to an eigen analysis problem in the wave coefficient space. Next, the process of the optimization is expressed using a Schrödinger-type equation, so global optimiza...

We study the modulational instability in surface gravity waves with random phase spectra. Starting from the nonlinear Schrödinger equation and using the Wigner-Moyal transform, we study the stability of the narrow-banded approximation of a typical wind-wave spectrum, i.e., the JONSWAP spectrum. By performing numerical simulations of the nonlinear Schrödinger equation we show that in the unstabl...

This paper derives a 2D, quasi-linear Schrödinger equation in Lagrangian coordinates that describes the effects of weak pressure gradients on large amplitude inertial oscillations in a rotating shallow fluid. The coefficients of the equation are singular at values of the gradient of the wave amplitude that correspond to the vanishing of the Jacobian of the transformation from Lagrangian to Eule...

Linear stability analysis of pulses is considered in this review chapter. The Evans function is an analytic tool whose zeros correspond to eigenvalues. Herein, the general manner of its construction shown. Furthermore, the construction is done explicitly for the linearization of the nonlinear Schrödinger equation about the 1-soliton solution. In an explicit calculation, it is shown how the Evan...

Singularly perturbed vector nonlinear Schrödinger equations (PVNLS) are investigated. Emphasis is placed upon the relation with their restriction: The singularly perturbed scalar nonlinear Schrödinger equation (PNLS) studied in [10]. It turns out that the persistent homoclinic orbit for the PNLS [10] is the only one for the PVNLS, asymptotic to the same saddle.

Cubic Schrödinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed. It is shown that along both, the solution of the nonlinear Schrödinger equation as well as the solution of the semi-discretized equation, the actions of the linear Schrödinger equation are approximately conserved over long times. This also allows to show app...

Nonlinear Schrödinger equation, complemented by a confining potential, possesses a discrete set of stationary solutions. These are called coherent modes, since the nonlinear Schrödinger equation describes coherent states. Such modes are also named topological because the solutions corresponding to different spectral levels have principally different spatial dependences. The theory of resonant e...