The nonlinear Schrödinger equation (NLS) is a canonical evolution equation, which describes the dynamics of weakly nonlinear wave packets in time and space in a wide range of physical media, such as nonlinear optics, cold gases, plasmas and hydrodynamics. Due to its integrability, the NLS provides families of exact solutions describing the dynamics of localised structures which can be observed ...

In the framework of the continuum approximation, localized modes in nonlinear lattices ~‘‘intrinsic localized modes’’ or ‘‘discrete breathers’’! are described by the nonlinear Schrödinger ~NLS! equation. We go beyond this approximation and analyze what kind of qualitatively new effects can be introduced by discreteness. Taking into account the higher-order linear and nonlinear dispersion terms ...

We consider the minimal mass m0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + ∆u = μ|u|4/du to blow up. If m0 is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in Lx(R d) is compact after quotienting out by the symmetries of the equation. A similar res...

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [4] to be linearly stable and the Floquet–Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are or...

Three different types of chaotic behavior and instabilities (homoclinic chaos, hyperbolic resonance, and parabolic resonance) in Hamiltonian perturbations of the nonlinear Schrödinger (NLS) equation are described. The analysis is performed on a truncated model using a novel framework in which a hierarchy of bifurcations is constructed. It is demonstrated numerically that the forced NLS equation...

In this paper, we give new solutions of the focusing NLS equation as a quotient of two determinants. This formulation gives in the case of the order 3, new deformations of the Peregrine breather with four parameters. This gives a very efficient procedure to construct families of quasirational solutions of the NLS equation and to describe the apparition of multi rogue waves. With this method, we...

A regularized α−system of the Nonlinear Schrödinger Equation (NLS) with 2σ nonlinear power in dimension N is studied. We prove existence and uniqueness of local solution in the case 1 ≤ σ < 4 N−2 and existence and uniqueness of global solution in the case 1 ≤ σ < 4 N . When α → 0, this regularized system will converge to the classical NLS in the appropriate range. In particular, the purpose of ...

We study the conditions for soliton-like wave propagation in the Photorefractive (PR) and electro-optic (i.e., Pockels) material, by using Nonlinear Schrodinger (NLS) equation. The complete NLS equation is solved analytically and numerically by transforming it into the phase space. Our results clearly show the existence of both the dark and bright solitary solutions for the PR medium. Interesti...

It is shown that the capacity of the channel modeled by (a discretized version of) the stochastic nonlinear Schrödinger (NLS) equation is upper-bounded by logp1` SNRq with SNR “ P0{σ pzq, where P0 is the average input signal power and σpzq is the total noise power up to distance z. This implies that the capacity of this model of the single-user optical fiber channel is below the capacity of the...