Related papers: The bidirectional NLS approximation for the one-di…
In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic Nonlinear Schrodinger equation (NLS) on the real line R and on the circle T for solutions below the L^2-threshold. We point out common…
In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham…
We construct in this paper global (for $t \geq 0$) and bounded solutions $u(t)$ for the nonlinear Schr\"odinger equation \[i \partial_t u + \Delta u + |u|^{p-1} u = 0, \quad t \in \mathbb{R}, x \in \mathbb{R}^d\] in mass sub-critical cases…
We study both analytically and numerically the nonlinear stage of the instability of one-dimensional solitons in a small vicinity of the transition point from supercritical to subcritical bifurcations in the framework of the generalized…
We present trapped solitary wave solutions of a coupled nonlinear Schr\"odinger system in $1$+$1$ dimensions in the presence of an external, supersymmetric and complex $\mathcal{PT}$-symmetric potential. The Schr\"odinger system this work…
We study the evolution of localized wave groups in unidirectional water wave envelope equations (nonlinear Schrodinger (NLS) and modified NLS (MNLS)). These localizations of energy can lead to disastrous extreme responses (rogue waves).…
We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schr\"odinger type and have recently been obtained in \cite{DLS}…
It has been suggested that the nonlinear Schr\"odinger-Newton equation might approximate the coupling of quantum mechanics with gravitation, particularly in the context of the M{\o}ller-Rosenfeld semiclassical theory. Numerical results for…
A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$…
We solve the (2+1)D nonlinear Helmholtz equation (NLH) for input beams that collapse in the simpler NLS model. Thereby, we provide the first ever numerical evidence that nonparaxiality and backscattering can arrest the collapse. We also…
We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schr\"odinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it…
A fundamental two-fluid model for describing dynamics of a plasma is the Euler-Poisson system, in which compressible ion and electron fluids interact with their self-consistent electrostatic force. Global smooth electron dynamics were…
Given constant data of density $\rho_0$, velocity $-u_0{\bf e}_r$, pressure $p_0$ and electric force $-E_0{\bf e}_r$ for supersonic flow at the entrance, and constant pressure $p_{\rm ex}$ for subsonic flow at the exit, we prove that…
One of the most profound questions of mathematical physics is that of establishing from first principles the hydrodynamic equations in large, isolated, strongly interacting many-body systems. This involves understanding relaxation at long…
We consider semilinear hyperbolic systems with a trilinear nonlinearity. Both the differential equation and the initial data contain the inverse of a small parameter $\varepsilon$, and typical solutions oscillate with frequency proportional…
We apply the idea of using a combination of virial inequalities and Kato smoothing, previously applied to NLS and generalized KdV pure power equations to Euler-Poisson: we assume that a solution remains very close for all times to a soliton…
We prove the existence of travelling-wave solutions for a system of coupled nonlinear Schr\"{o}dinger equations arising in nonlinear optics. Such a system describes second-harmonic generation in optical materials with $\chi^{(2)}$…
We introduce spatiotemporal optical dark X solitary waves of the (2+1)D {hyperbolic} nonlinear Schr\"odinger equation (NLSE), that rules wave propagation in a self-focusing and normally dispersive medium. These analytical solutions are…
We study various properties of solutions of an extended nonlinear Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric evolution problems -- including vortex filament dynamics -- and governs propagation of short pulses…
In this paper, we consider the dispersive limit of the Euler-Poisson system for ion-acoustic waves. We establish that under the Gardner-Morikawa type transformations, the solutions of the Euler-Poisson system converge globally to the…