scholarly journals The Schrödinger equation with spatial white noise potential

2018 ◽  
Vol 23 (0) ◽  
Author(s):  
Arnaud Debussche ◽  
Hendrik Weber
Keyword(s):  
Schrödinger Equation ◽  
White Noise ◽  
Schrodinger Equation

Behaviour of solutions to the 1D focusing stochastic L2-critical and supercritical nonlinear Schrödinger equation with space-time white noise

2021 ◽  
Author(s):  
Annie Millet ◽  
Svetlana Roudenko ◽  
Kai Yang
Keyword(s):  
Schrödinger Equation ◽  
White Noise ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Random Variable ◽  
Space Time ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Deterministic Setting

Abstract We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that, we study the influence of noise on the global behaviour of solutions. In particular, we show that the noise may induce blow up, thus ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow up happens, it has the same dynamics as in the deterministic setting; however, there is a (random) shift of the blow-up centre, which can be described as a random variable normally distributed.

White noise in the two-dimensional nonlinear schrödinger equation

1995 ◽  
Vol 57 (1-2) ◽  
pp. 3-15 ◽  
Author(s):  
O. Bang ◽  
P.L. Christiansen ◽  
F. If2 ◽  
K.⊘. Rasmussen ◽  
Y.B. Gaididei
Keyword(s):  
Schrödinger Equation ◽  
White Noise ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Two Dimensional ◽  
Nonlinear Schrödinger

scholarly journals Collapse of solitary excitations in the nonlinear Schrödinger equation with nonlinear damping and white noise

1996 ◽  
Vol 54 (1) ◽  
pp. 924-930 ◽  
Author(s):  
Peter L. Christiansen ◽  
Yuri B. Gaididei ◽  
Magnus Johansson ◽  
Kim Ø. Rasmussen ◽  
Irina I. Yakimenko
Keyword(s):  
Schrödinger Equation ◽  
White Noise ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Damping ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

scholarly journals STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION WITH ALMOST SPACE–TIME WHITE NOISE

2019 ◽  
Vol 109 (1) ◽  
pp. 44-67 ◽  
Author(s):  
JUSTIN FORLANO ◽  
TADAHIRO OH ◽  
YUZHAO WANG
Keyword(s):  
Schrödinger Equation ◽  
White Noise ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Time ◽  
Nonlinear Schrodinger Equation ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
The One ◽  
Cubic Nonlinear Schrödinger Equation

We study the stochastic cubic nonlinear Schrödinger 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 noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).

scholarly journals The 1D Schrödinger equation with a spacetime white noise: the average wave function

2019 ◽  
Vol 23 ◽  
pp. 338-349
Author(s):  
Yu Gu
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
White Noise ◽  
Effective Potential ◽  
Schrodinger Equation ◽  
Average Wave

For the 1D Schrödinger equation with a mollified spacetime white noise, we show that the average wave function converges to the Schrödinger equation with an effective potential after an appropriate renormalization.

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