Related papers: Two dimensional nonlinear Schr\"odinger equation w…
We study the scattering behavior of global solutions to stochastic nonlinear Schr\"odinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is…
We prove global existence for the one-dimensional cubic non-linear Schr\"odinger equation in modulation spaces $M_{p,p'}$ for $p$ sufficiently close to $2$. In contrast to known results, our result requires no smallness condition on initial…
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 prove symplectic non-squeezing for the cubic nonlinear Schr\"odinger equation on the line via finite-dimensional approximation.
We provide the rigorous justification of the NLS approximation, in Sobolev regularity, for a class of quasilinear Hamiltonian Klein Gordon equations with quadratic nonlinearities on large one-dimensional tori $\T_L:=\mathbb{R}/(2\pi L…
We prove that solutions to non-linear Schr\"odinger equations in two dimensions and in the exterior of a bounded and smooth star-shaped obstacle scatter in the energy space. The non-linear potential is defocusing and grows at least as the…
Persistence of stationary and traveling single-humped localized solutions in the spatial discretizations of the nonlinear Schrodinger (NLS) equation is addressed. The discrete NLS equation with the most general cubic polynomial function is…
We derive a class of discrete nonlinear Schr{\"o}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of…
We study the problem of unconditional uniqueness of solutions to the cubic nonlinear Schr\"odinger equation. We introduce a new strategy to approach this problem on bounded domains, in particular on rectangular tori. It is a known fact that…
We consider the nonlinear Schr\"odinger equation on the $d$-dimensional torus $\mathbb T^d$, with the nonlinearity of polynomial type $|u|^{2\sigma}u$. For any $\sigma \in \mathbb N$ and $s>\frac d2$ we prove that adding to this equation a…
The purpose of this article is to construct global solutions, in a probabilistic sense, for the nonlinear Schr{\"o}dinger equation posed on $\mathbb{R}^d$, in a supercritical regime. Firstly, we establish Bourgain type bilinear estimates…
We prove the existence of a 2-parameter family of small quasi-periodic in time solutions of discrete nonlinear Schr\"odinger equation (DNLS). We further show that all small solutions of DNLS decouples to this quasi-periodic solution and…
Global existence and scattering for the nonlinear defocusing Schr\"odinger equation in 2 dimensions are known for domains exterior to star-shaped obstacles and for nonlinearities that grow at least as the quintic power. In this paper, we…
In this paper we prove global existence and uniqueness of solutions to the stochastic logarithmic Schr\"odinger equation with linear multiplicative noise. Our approach is mainly based on the rescaling approach and the method of maximal…
In this paper, we consider solutions to the following fourth order anisotropic nonlinear Schr\"odinger equation in $\R \times \R^2$, $$ \left\{ \begin{aligned} &\textnormal{i}\partial_t\psi+\partial_{xx} \psi-\partial_{yyyy} \psi…
It is shown that using the similarity transformations, a set of three-dimensional p-q nonlinear Schrodinger (NLS) equations with inhomogeneous coefficients can be reduced to one-dimensional stationary NLS equation with constant or varying…
For nonlinear dispersive systems, the nonlinear Schr\"odinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally…
We study the one dimensional nonlinear Schr\"odinger equation with power nonlinearity $|u|^{\alpha - 1} u$ for $\alpha \in [1,5]$ and initial data $u_0 \in L^2(\mathbb{R}) + H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy…
We are concerned with the two-power nonlinear Schr\"odinger-type equations with non-local terms. We consider the framework of Sobolev-Lorentz spaces which contain singular functions with infinite-energy. Our results include global…
We study the stochastic viscous nonlinear wave equations (SvNLW) on $\mathbb T^2$, forced by a fractional derivative of the space-time white noise $\xi$. In particular, we consider SvNLW with the singular additive forcing $D^\frac{1}{2}\xi$…