Related papers: Localized States in Discrete Nonlinear Schr\"{o}di…
We consider the localized modes (bright solitons) described by one-dimensional quintic nonlinear Schrodinger equation with a periodic potential. In the case of attractive nonlinearity we deduce sufficient conditions for collapse. We show…
We study the one-dimensional nonlinear Schr\"odinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a…
The creation of stable 1D and 2D localized modes in lossy nonlinear media is a fundamental problem in optics and plasmonics. This article gives a short review of theoretical methods elaborated for this purpose, using localized gain applied…
We study a damped stochastic non-linear Schr\"{o}dinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markovian transition semi-group toward a…
We consider a class of nonlinear Schroedinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$)…
We consider an array of double oligomers in an optical waveguide device. A mathematical model for the system is the coupled discrete nonlinear Schr\"odinger (NLS) equations, where the gain-and-loss parameter contributes to the…
We construct multiple families of solitary standing waves of the discrete cubically nonlinear Schr\"{o}dinger equation (DNLS) in dimensions $d=1,2$ and $3$. These states are obtained via a bifurcation analysis about the continuum (NLS)…
We study localized two- and three-dimensional Langmuir solitons in the framework of model based on generalized nonlinear Schr\"odinger equation that accounts for local and nonlocal contributions to electron-electron nonlinearity. General…
We study the formation of stationary localized states using the discrete nonlinear Schr\"{o}dinger equation in a Cayley tree with connectivity $K$. Two cases, namely, a dimeric power law nonlinear impurity and a fully nonlinear system are…
A discrete version of the inverse scattering method proposed by Ablowitz and Ladik is generalized to study an integrable full-discretization (discrete time and discrete space) of the coupled nonlinear Schr\"{o}dinger equations. The…
We study how dark solitons, i.e. solutions of one-dimensional single-particle nonlinear time-dependent Schr\"odinger equation, emerge from eigenstates of a linear many-body model of contact interacting bosons moving on a ring, the…
We study the existence and stability of localized states in the two-dimensional (2D) nonlinear Schrodinger (NLS)/Gross-Pitaevskii equation with a symmetric four-well potential. Using a fourmode approximation, we are able to trace the…
We give a new proof of the scattering below the ground state energy level for a class of nonlinear Schr\"odinger equations (NLS) with mass-energy intercritical competing nonlinearities. Specifically, the NLS has a focusing leading order…
We consider the continuous resonant (CR) system of the 2D cubic nonlinear Schr{\"o}dinger (NLS) equation. This system arises in numerous instances as an effective equation for the long-time dynamics of NLS in confined regimes (e.g. on a…
We consider numerical instability that can be observed in simulations of localized solutions of the generalized nonlinear Schr\"odinger equation (NLS) by a split-step method where the linear part of the evolution is solved by a…
We overview the properties of nonlinear guided waves and (bright and dark) spatial optical solitons in a periodic medium created by a sequence of linear and nonlinear layers. First, we consider a single layer with a cubic nonlinear response…
It is known that multidimensional complex potentials obeying $\mathcal{PT}$-symmetry may possess all real spectra and continuous families of solitons. Recently it was shown that for multi-dimensional systems these features can persist when…
We report that infinite and semi-infinite lattices with spatially inhomogeneous self-defocusing (SDF)\ onsite nonlinearity, whose strength increases rapidly enough toward the lattice periphery, support stable unstaggered (UnST) discrete…
With the stationary solution assumption, we establish the connection between the nonlocal nonlinear Schr\"{o}dinger (NNLS) equation and an elliptic equation. Then, we obtain the general stationary solutions and discuss the relevance of…
We investigate the existence of multiple bound states of prescribed mass for the nonlinear Schr\"odinger equation on a noncompact metric graph. The main feature is that the nonlinearity is localized only in a compact part of the graph. Our…