Related papers: Dam breaks in the discrete nonlinear Schr\"odinger…
We study the nonlinear interactions of waves with a doubled-peaked power spectrum in shallow water. The starting point is the prototypical equation for nonlinear uni-directional waves in shallow water, i.e. the Korteweg de Vries equation.…
We provide the rigorous derivation of the wave kinetic equation from the cubic nonlinear Schr\"odinger (NLS) equation at the kinetic timescale, under a particular scaling law that describes the limiting process. This solves a main…
We analyze the existence and stability of localized solutions in the one-dimensional discrete nonlinear Schr\"{o}dinger (DNLS) equation with a combination of competing self-focusing cubic and defocusing quintic onsite nonlinearities. We…
We derive a class of discrete nonlinear Schr{\"o}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of…
We study here the nonlinear Schrodinger Equation (NLS) as the first term in a sequence of approximations for an electromagnetic (EM) wave propagating according to the nonlinear Maxwell equations (NLM). The dielectric medium is assumed to be…
Dynamics of interacting cold atomic gases have recently become a focus of both experimental and theoretical studies. Often cold atom systems show hydrodynamic behavior and support the propagation of nonlinear dispersive waves. Although this…
We present a model of optical communication system for high-bit-rate data transmission in the nonreturn-to-zero (NRZ) format over transoceanic distance. The system operates in a small group velocity dispersion regime, and the model equation…
For nonlinear hyperbolic systems of conservation laws in one space variable, we establish the existence of nonclassical entropy solutions exhibiting nonlinear interactions between shock waves with strong strength. The proposed theory is…
The spatiotemporal complexity induced by perturbed initial excitations through the development of modulational instability in nonlinear lattices with or without disorder, may lead to the formation of very high amplitude, localized transient…
We suggest a method for calculation of parameters of dispersive shock waves in framework of Whitham modulation theory applied to non-integrable wave equations with a wide class of initial conditions corresponding to propagation of a pulse…
We consider the initial value problem for the inhomogeneous nonlinear Schr\"odinger equation with double nonlinearities (DINLS) \begin{equation*} i \partial_t u + \Delta u = \lambda_1 |x|^{-b_1}|u|^{p_1}u +…
The defocusing nonlinear Schr\"odinger (NLS) equation has no the modulational instability, and was not found to possess the rogue wave (RW) phenomenon up to now. In this paper, we firstly investigate some novel nonlinear wave structures in…
We consider the discrete solitons bifurcating from the anti-continuum limit of the discrete nonlinear Schr\"{o}dinger (NLS) lattice. The discrete soliton in the anti-continuum limit represents an arbitrary finite superposition of {\em…
We consider a scaling limit of a nonlinear Schr\"odinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized…
In this paper, we investigate the damped stochastic nonlinear Schr\"odinger(NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of the damped…
We consider several different bidirectional Whitham equations that have recently appeared in the literature. Each of these models combine the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow…
Dispersive shock waves (DSW) are a salient feature of long water waves often observed in tidal bores and tsunami/meteotsunami contexts. Their interaction with bathymetry is poorly understood. The shoreline hazard from tsunamis and…
We investigate a class of nonlinear equations of Schr\"odinger type with competing inhomogeneous nonlinearities in the non-radial inter-critical regime, \begin{align*} i \partial_t u +\Delta u &=|x|^{-b_1} |u|^{p_1-2} u - |x|^{-b_2}…
The discrete self-trapping equation (DST) represents an useful model for several properties of one-dimensional nonlinear molecular crystals. The modulational instability of DST equation is discussed from a statistical point of view,…
The small dispersion limit of the focusing nonlinear Schro\"odinger equation (NLS) exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales. The non self-adjoint scattering problem…