Related papers: The NLS equation in dimension one with spatially c…
We consider a scaling limit of a nonlinear Schr\"odinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized…
We consider the 1D nonlinear Schr\"odinger equation (NLS) with focusing point nonlinearity, $$ (\delta\text{NLS}) \qquad i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, $$ where $\delta=\delta(x)$ is the delta function…
A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$…
We consider the cubic nonlinear Schr\"odinger (NLS) equation set on a two dimensional box of size $L$ with periodic boundary conditions. By taking the large box limit $L \to \infty$ in the weakly nonlinear regime (characterized by smallness…
We consider the long-time dynamics of focusing energy-critical Schr\"odinger equation perturbed by the $\dot{H}^\frac{1}{2}$-critical nonlinearity and with inverse-square potential(CNLS$_a$) in dimensions $d\in\{3,4,5\}$…
We study the nonlinear-damping continuation of singular solutions of the critical and supercritical NLS. Our simulations suggest that for generic initial conditions that lead to collapse in the undamped NLS, the solution of the…
In this study, we consider the nonlinear Sch\"odinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean…
We consider the $L^{2}$-critical quintic focusing nonlinear Schr\"odinger equation (NLS) on ${\bf R}$. It is well known that $H^{1}$ solutions of the aforementioned equation blow up in finite time. In higher dimensions, for $H^{1}$…
We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…
We consider discrete nonlinear Schr\"odinger equations (DNLS) on the lattice $h\mathbb{Z}^d$ whose linear part is determined by the discrete Laplacian which accounts only for nearest neighbor interactions, or by its fractional power. We…
We consider nonlinear dispersive equations of Schr\"odinger-type involving fractional powers $0<s\le 1$ of the Laplacian and a defocusing power-law nonlinearity. We conduct numerical simulations in the case of small, energy supercritical…
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…
The fractional discrete nonlinear Schr\"odinger equation (fDNLS) is studied on a periodic lattice from the analytic and dynamic perspective by varying the mesh size $h>0$ and the nonlocal L\'evy index $\alpha \in (0,2]$. We show that the…
The initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for a higher order nonlinear Schr\"odinger (HNLS) equation is considered. Existence and uniqueness of global weak solutions…
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…
In this work we deal with a following nonlinear Schrodinger equation in dimension greater or equal to 3, with a subcritical power-type nonlinearity and a positive potential satisfying a local condition. We prove the existence and…
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 the nonlinear Schr\"odinger equation with focusing quintic and defocusing cubic nonlinearity in three space dimensions: \[ (i\partial_t+\Delta)u = |u|^2 u - |u|^4 u. \] In [18, 23], the authors classified the dynamics of…
The paper discusses nonlinear singular perturbations of delta type of the fractional Schr\"odinger equation $\imath\partial_t\psi=\left(-\triangle\right)^s\psi$, with $s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local…
We consider the cubic Nonlinear Schroedinger Equation (NLS) in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a-priori local in time Hs bounds in terms of the Hs size of the initial data for s >=-1/4…