Related papers: Global Analytic Solutions for the Nonlinear Schr\"…
We present basic results, known and new, on nontrivial solutions of periodic stationary nonlinear Schr\"odinger equations. We also sketch an application to nonlinear optics and discuss some open problems.
We study a system of inhomogeneous nonlinear Schr\"odinger equations that emerge in optical media with a $\chi^{(2)}$ nonlinearity. This nonlinearity, whose local strength is subject to a cusp-shaped spatial modulation, $\chi^{(2)}\sim…
We consider the initial value problem for the fractional nonlinear Schr\"odinger equation with a fractional dissipation. Global existence and scattering are proved depending on the order of the fractional dissipation.
In this paper the one-dimensional nonparaxial nonlinear Schr\"odinger equation is considered. This was proposed as an alternative to the classical nonlinear Schr\"odinger equation in those situations where the assumption of paraxiality may…
In this paper we prove global existence and global behavior of solutions to quasilinear wave-Klein-Gordon systems in $\mathbb{R}^{1+2}$ with quadratic nonlinearities satisfying the null condition. We consider small, regular and compactly…
In this work, we mainly focus on the energy-supercritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= \mu|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $\mu=\pm1$ and $p>\frac4{d-2}$. %In this work, we consider the…
The purpose of this article is to construct global solutions for some super-crtical Schrodinger equations using the theory of random data introduced by N.Burq and N.Tzvetkov. We begin our study by the cubic equation in three dimension.…
This paper is dedicated to the study of the derivative nonlinear Schr\"odinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global…
Consider, in dimension 3, a system of coupled Klein-Gordon equations with different speeds, and an arbitrary quadratic nonlinearity. We show, for data which are small, smooth, and localized, that a global solution exists, and that it…
We consider the semi-classical limit of nonlinear Schrodinger equations in the presence of both a polynomial nonlinearity and thederivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not…
We consider the dispersion managed nonlinear Schr\"dinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in $L_x^2$ for small initial data employing…
A procedure of solving nonstationary Schredinger equations in the exact analytic form is elaborated on the basis of exactly solvable stationary models. The exact solutions are employed to study the nonadiabatic geometric phase.
We consider the final-data problem for systems of nonlinear Schr\"odinger equations with $L^2$ subcritical nonlinearity. An asymptotically free solution is uniquely obtained for almost every randomized asymptotic profile in…
We investigate the Cauchy problem for the nonlinear Schr\"odinger equation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global…
For the Schr\"odinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and…
We study 1D NLS with non-gauge invariant quadratic nonlinearity on the torus. The Cauchy problem admits trivial global solutions which are constant with respect to space. The non-existence of global solutions also has been studied only by…
Using perturbative methods, we analyse a nonlinear generalisation of Schrodinger's equation that had previously been obtained through information-theoretic arguments. We obtain analytical expressions for the leading correction, in terms of…
This article is devoted to a general class of one dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and…
We study the wave front set of the solutions of the initial value problem for nonlinear Schr\"{o}dinger equations via wave packet transform. We give an sufficient condition which assures that the solutions is in Sobolev space of order s in…
We consider the existence of normalized solutions to nonlinear Schr\"odinger equations on noncompact metric graphs in the $L^2$ supercritical regime. For sufficiently small prescribed mass ($L^2$ norm), we prove existence of positive…