Abstract We justify rigorously the convergence of the amplitude of solutions of Nonlinear-Schrödinger type Equations with non zero limit at infinity to an asymptotic regime governed by the Kortewegde Vries equation in dimension 1 and the Kadomtsev-Petviashvili I equation in dimensions 2 and more. We get two types of results. In the one-dimensional case, we prove directly by energy bounds that t...

We investigate wave collapse ruled by the generalized nonlinear Schrödinger (NLS) equation in 1+1 dimensions, for localized excitations with non-zero background, establishing through virial identities a new criterion for blow-up. When collapse is arrested, a semiclassical approach allows us to show that the system can favor the formation of dispersive shock waves. The general findings are illus...

We consider existence and stability of dispersion-managed solitons in the two approximations of the periodic nonlinear Schrödinger (NLS) equation: (i) a dynamical system for a Gaussian pulse and (ii) an average integral NLS equation. We apply normal form transformations for finite-dimensional and infinite-dimensional Hamiltonian systems with periodic coefficients. Firstorder corrections to the ...

Two discretizations of the vector nonlinear Schrödinger (NLS) equation are studied. One of these discretizations, referred to as the symmetric system, is a natural vector extension of the scalar integrable discrete NLS equation. The other discretization, referred to as the asymmetric system, has an associated linear scattering pair. General formulae for soliton solutions of the asymmetric syste...

We study the evolution of the one dimensional periodic cubic Schrödinger equation (NLS) with bounded variation data. For the linear evolution, it is known that for irrational times the solution is a continuous, nowhere differentiable fractal-like curve. For rational times the solution is a linear combination of finitely many translates of the initial data. Such a dichotomy was fist observed by ...

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...

Interactions of wavelength-division multiplexed (WDM) dispersion-managed solitons solitons are studied by means of numerical simulations of both the nonlinear Schrödinger (NLS) equation and the nonlocal equation of the NLS type that describes pulse dynamics in systems with strong dispersion management. The interaction properties are found to depend significantly on the values of system paramete...

In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, darkand Peregrine-type soliton solutio...

Deep-water, narrow-banded wavetrains of uniform amplitude propagating in one horizontal dimension were shown to be unstable to small perturbations with nearly the same frequency and direction about 40 years ago. More recently, interactions of two narrow-banded wavetrains of uniform amplitude propagating in arbitrary directions were shown to be similarly unstable. In both cases, the instabilitie...