Related papers: Normal Form for the Schr\"odinger equation with an…
We study the quintic Non Linear Schr\"odinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in…
We investigate the existence, non-existence, and multiplicity of positive solutions to a class of quasilinear Schrodinger equations with a prescribed mass condition in higher dimensions. Using the dual approach, the equation is transformed…
This paper is devoted to the analysis of the continuous resonant (CR) equation, in dimensions greater than 2. This equation arises as the large box (or high frequency) limit of the nonlinear Schrodinger equation on the torus, and was…
We consider Darboux transformations for the derivative nonlinear Schr\"odinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant…
We prove the existence of non-trivial solutions to a system of coupled, nonlinear, Schroedinger equations with general nonlinearity.
Nonlinear Schr\"odinger equation, complemented by a confining potential, possesses a discrete set of stationary solutions. These are called coherent modes, since the nonlinear Schr\"odinger equation describes coherent states. Such modes are…
We consider a finite but arbitrarily large Klein-Gordon chain, with periodic boundary conditions. In the limit of small couplings in the nearest neighbor interaction, and small (total or specific) energy, a high order resonant normal form…
In this paper, we study the local well-posedness of nonlinear Schr\"odinger equations on tori $\mathbb{T}^{d}$ at the critical regularity. We focus on cases where the nonlinearity $|u|^{a}u$ is non-algebraic with small $a>0$. We prove the…
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 deformation of the nonlinear Schr\"odinger equation recently derived in the context of deformation of hierarchies of integrable systems. This systematic method also led to known integrable equations such as the Camassa-Holm…
This paper is concerned with the derivative nonlinear Schr\"{o}dinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small…
We examine a recently-proposed family of nonlinear Schr\"odinger equations [J. Phys. A: Math. Gen. 27:1771(1994)] with respect to a group of transformations that linearize a subfamily of them. We investigate the structure of the whole…
We study the nonlinear Schr\"odinger equation with a competing cubic-quintic power law nonlinearity on the waveguide domain $\mathbb R_x \times \mathbb T_{L_y}$. This model is globally well-posed and admits line solitary wave solutions,…
We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results…
We consider the nonlinear Schr\"odinger equation with a focusing cubic term and a defocusing quintic nonlinearity in dimensions two and three. The core of this article is the notion of stability of solitary waves. We recall the two standard…
We prove a Nekhoroshev type theorem for the nonlinear Schr\"odinger equation $$ iu_t=-\Delta u+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$ where $V$ is a typical smooth potential and $g$ is analytic in both variables. More…
We obtain novel nonlinear Schr\"{o}dinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
We consider NLS on $\T^2$ with multiplicative spatial white noise and nonlinearity between cubic and quartic. We prove global existence, uniqueness and convergence almost surely of solutions to a family of properly regularized and…
We consider the radial nonlinear Schr\"odinger equation $i\partial_tu +\Delta u = |u|^{p-1}u$ in dimension $d\geqslant 2$ for $p\in \left(1,1+\frac{4}{d}\right]$ and construct a natural Gaussian measure $\mu_0$ which support is almost…