Related papers: Global existence for the defocusing nonlinear Schr…

We consider the Cauchy problem of nonlinear Schr\"odinger equations (NLS) with almost periodic functions as initial data. We first prove that, given a frequency set $\pmb{\omega} =\{\omega_j\}_{j = 1}^\infty$, NLS is local well-posed in the…

The spatially periodic initial problem and Cauchy problem for nonlinear Schr\"odinger equations are considered. The existence and uniqueness of global solution with infinite smooth initial data $u_0$, i.e. $u_0,\;|u_0|^{2p}u_0\in…

This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schr\"odinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency…

We study the nonlinear Schr\"odinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially…

We study the Cauchy problem for the defocusing nonlinear Schr\"odinger (NLS) equation under the assumption that the solution vanishes as $x \to + \infty$ and approaches an oscillatory plane wave as $x \to -\infty$. We first develop an…

This is the second part of a two-paper series studying the nonlinear Schr\"odinger equation with quasi-periodic initial data. In this paper, we focus on the quasi-periodic Cauchy problem for the derivative nonlinear Schr\"odinger equation.…

In this paper, we address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schr\"{o}dinger (nonlocal NLS) equation with the initial data $q_0(x)\in H^{1,1}(\R)$ with the $L^1(\R)$ small-norm…

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…

We consider the Cauchy problem for the defocusing stochastic nonlinear Schr\"odinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in…

We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear Schr\"odinger equation (NLS) with initial data below L^2. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove…

This article is concerned with the small data problem for the cubic nonlinear Schr\"odinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simpler approach in order to prove that global…

We consider solutions of the defocusing nonlinear Schr\"odinger (NLS) equation on the half-line whose Dirichlet and Neumann boundary values become periodic for sufficiently large $t$. We prove a theorem which, modulo certain assumptions,…

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…

We study the Cauchy problem for the cubic nonlinear Schroedinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we…

We consider the Cauchy problem to the 3D fractional Schr\"odinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty…

We establish global existence for the energy-critical nonlinear Schr\"odinger equation on $\mathbb{S}^3$. This follows similar lines to the work on $\mathbb{T}^3$ but requires new extinction results for linear solutions and bounds on the…

We prove the existence of global analytic solutions to the nonlinear Schr\"odinger equation in one dimension for a certain type of analytic initial data in $L^2$.

We consider the Cauchy problem for the (derivative) nonlocal NLS in super-critical function spaces $E^s_\sigma$ for which the norms are defined by $$ \|f\|_{E^s_\sigma} = \|\langle\xi\rangle^\sigma 2^{s|\xi|}\hat{f}(\xi)\|_{L^2}, \ s<0, \…

We address the open problem of determining which classes of time-dependent linear Schr\"odinger equations and focusing and defocusing cubic and quintic non-linear Schr\"odinger equations (NLS) on unbounded domains that can be computed by an…

We consider the Cauchy problem for (energy-subcritical) nonlinear Schr\"odinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum…