Related papers: Decay estimates for nonlinear Schr\"odinger equati…
The initial value problem is considered for a higher order nonlinear Schr\"odinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional…
We establish dispersive estimates and local decay estimates for the time evolution of non-self-adjoint matrix Schr\"odinger operators with threshold resonances in one space dimension. In particular, we show that the decay rates in the…
This paper is concerned with the $L^{2}$-decay estimate of solutions to nonlinear dissipative Schr\"odinger equations with power-type nonlinearity of the order $p$. It is known that the sign of the real part of the dissipation coefficient…
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 present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schr\"odinger type and have recently been obtained in \cite{DLS}…
We establish Strichartz estimates (both reversed and some direct ones), pointwise decay estimates, and weighted decay estimates for the linear wave equation in dimension two with an almost scaling-critical potential, in the case when there…
In this paper, we prove the decay and scattering in the energy space for nonlinear Schr\"odinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. We will prove decay…
An explicit lifespan estimate is presented for the derivative Schr\"odinger equations with periodic boundary condition.
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem…
We consider the problem of recovering a spatially-localized cubic nonlinearity in a nonlinear Schr\"odinger equation in dimensions two and three. We prove that solutions with data given by small-amplitude wave packets accrue a nonlinear…
Consider the defocusing quintic nonlinear Schr\"{o}dinger equation on $\mathbf{R}^3$ with initial data in the energy space. This problem is "energy-critical" in view of a certain scale-invariance, which is a main source of difficulty in the…
In this paper we compute the higher order long time asymptotics of the defocussing nonlinear Schr\"odinger equation using the $\overline{\partial}$-nonlinear steepest descent method. We assume initial condition in weighted Sobolev space…
We analize the Nonlinear Schr\"odinger Equation.
The decay rates and spectroscopy of the $D$ and $D_s$ mesons are computed in a nonrelativistic phenomenological quark-antiquark potential of the type $V(r)=-{4/3}\frac{\alpha_s}{r}+A r^{\nu}$ with different choices of $\nu$. Numerical…
We present a new method for obtaining sharp asymptotics of solutions of the defocussing nonlinear Schr\"odinger (NLS) equation, based on dbar methods and under essentially minimal regularity assumptions on initial data.
The method of solving of nonlinear Schr\"odinger equation is considered. Some examples of its applications are demonstrated.
We give a survey on recent developments on nonlinear Schr\"odinger equations with dissipative structure based on the authors' recent works.
Consider the initial value problem for systems of cubic derivative nonlinear Schr\"odinger equations in one space dimension with the masses satisfying a suitable resonance relation. We give structural conditions on the nonlinearity under…
In this note, we generalize the nonlinearity-recovery result in [7] for classical cubic nonlinear Schr\"odinger equations to higher-order Schr\"odinger equations with a more general nonlinearity. More precisely, we consider a…
We consider the nonlinear Schr\"odinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip, Oh, Pocovnicu, and Visan, who proved scattering…