相关论文: Modified scattering for the cubic Schr{\"o}dinger …
We consider a stochastic nonlinear defocusing Schr\"{o}dinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear…
The two-dimensional cubic nonlinear Schr\"{o}dinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. The coupled two-dimensional cubic nonlinear Schr\"{o}dinger equations are…
We establish modified scattering for solutions of the cubic NLS on the line with a repulsive inverse square potential and small localized data. The method is based on a comparison between the free and distorted Galilei vector fields and a…
We prove sharp $L^\infty$ decay and modified scattering for the Schr\"odinger-Bopp-Podolsky equation in $2$ and $3$ spatial dimensions with small initial data chosen from a weighted Sobolev space.
In this article, we show that the solution to defocusing cubic nonlinear Schr\"odinger equation (NLS) posed on the two-dimensional waveguide \begin{align*} i\partial_tu+\Delta_{\R\times\T}u=|u|^2u \end{align*} is globally well-posed in…
We study the small data scattering problem in critical spaces for the nonlinear Schr\"odinger equation (NLS) on waveguide manifolds. Our work is primarily inspired by the recent paper of Kwak and Kwon \cite{KwakKwon} that established the…
This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schr\"{o}dinger equations (NLS). The method is explicit, unconditionally stable and time transversal invariant.…
We consider the cubic nonlinear Schr\"odinger (NLS) equation with a linear damping on the one dimensional torus and we investigate the stability of some solitary wave profiles within the dissipative dynamics. The undamped cubic NLS equation…
In this paper, we consider the following three dimensional defocusing cubic nonlinear Schr\"odinger equation (NLS) with partial harmonic potential \begin{equation*}\tag{NLS} i\partial_t u + \left(\Delta_{\mathbb{R}^3 }-x^2 \right) u = |u|^2…
We analyze dynamical properties of the logarithmic Schr{\"o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a…
We consider the scattering transform for the Schr\"odinger equation with a singular potential and no bound states. Using the Riccati representation for real-valued potentials on the line, we obtain invertibility and Lipschitz continuity of…
We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein-Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to…
We study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schr{\"o}dinger equations. First, we give global variational characterizations of each of…
We study the nonlinear Schr\"odinger equation posed on product spaces $\mathbf R^n\times \mathcal M^k$, for $n\geq 1$ and $k\geq1$, with $\mathcal M^k$ any $k$-dimensional compact Riemaniann manifold. The main results concern global…
In this paper we study microlocal singularities of solutions to Schrodinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in…
The purpose of this paper is to establish a geometric scattering result for a conformally invariant nonlinear wave equation on an asymptotically simple spacetime. The scattering operator is obtained via trace operators at null infinities.…
We study the theory of scattering for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling,in space dimension 3.We prove in particular the existence of modified wave operators for that system with…
We consider the Cauchy problem for the defocusing cubic nonlinear Schr\"odinger equation (NLS) on the waveguide manifold $\mathbb{R}^3\times\mathbb{T}$ and establish almost sure scattering for random initial data, where no symmetry…
We obtain polynomial bounds on the growth in time of Sobolev norm of solutions to the cubic defocusing nonlinear Schrodinger equation on two dimensional product space. We also give the angular improved bilinear Strichartz estimates for…
We consider a nonlinear dispersive equation with a quasilinear quadratic term. We establish two results. First, we show that solutions to this equation with initial data of order $\mathcal{O}(\varepsilon)$ in Sobolev norms exist for a time…