相关论文: The logarithmic Schr{\"o}dinger equation with spat…
We consider the linear and nonlinear Schr{\"o}dinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in [8] and conserved…
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…
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…
In this paper, we prove existence and uniqueness of energy solutions for nonlinear Schr\"odinger equations with a multiplicative white noise on $R^d$ with $d\le3$. We rely on an exponential trans-form and conserved quantities for existence…
We consider the nonlinear Schr\"odinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced…
In this paper, we study the stochastic logrithmic Schr\"odinger equation with saturated nonlinear multiplicative L\'evy noise. The global well-posedness is established for the stochastic logrithmic Schr\"odinger equation in an appropriate…
We study the stochastic cubic nonlinear Schr\"odinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space-time white…
In this article, we study a $d$-dimensional stochastic quadratic nonlinear Schr\"{o}dinger equation (SNLS), driven by a fractional derivative (of order $-\alpha<0$) of a space-time white noise: $$\left\{ \begin{array}{l}i\partial_t u-\Delta…
We consider the Cauchy problem for the logarithmic Schr\"odinger equation and prove uniqueness of weak $H^s(\mathbb{R}^d)$ solutions for $s\in(0,1)$, which improves on the previous uniqueness result in $H^1(\mathbb{R}^d)$. The proof is…
In this paper, we establish the existence and uniqueness of solutions of stochastic nonlinear Schr\"{o}dinger equations with additive jump noise in $L^2(\mathbb{R}^d)$. Our results cover all either focusing or defocusing nonlinearity in the…
We here show how the methods recently applied by [DW16] to solve the stochastic nonlinear Schr\"odinger equation on $\mathbb{T}^2$ can be enhanced to yield solutions on $\mathbb{R}^2$ if the non-linearity is weak enough. We prove that the…
The Cauchy problem for the stochastic nonlinear Schr\"odinger equation with multiplicative noise is considered where the nonlinear term is of power type and the noise coefficients are purely imaginary numbers. The main purpose of this paper…
We give decay estimates of the solution to the linear Schr\"odinger equation in dimension $d \geq 3$ with a small noise which is white in time and colored in space. As a consequence, we also obtain certain asymptotic behaviour of the…
We study a stochastic Schr{\"o}dinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension less than 3. When the Hurst index is large enough, we prove local well-posedness of the problem using…
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…
In this paper, we prove the global existence and uniqueness of the solution of the stochastic logarithmic Schr\"odinger (SlogS) equation driven by additive noise or multiplicative noise. The key ingredient lies on the regularized stochastic…
We consider a stochastic nonlinear Schr\"odinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic NLS in $H^1$ on compact manifolds and bounded domains. We construct a…
We investigate the presence of localized analytical solutions of the Schr\"odinger equation with logarithm nonlinearity. After including inhomogeneities in the linear and nonlinear coefficients, we use similarity transformation to convert…
In this article, we improve the Strichartz estimates obtained in [12] for the Schr\"odinger equation with white noise dispersion in one dimension. This allows us to prove global well posedness when a quintic critical nonlinearity is added…
For the 1D Schr\"odinger equation with a mollified spacetime white noise, we show that the average wave function converges to the Schr\"odinger equation with an effective potential after an appropriate renormalization.