scholarly journals Nonlinear Schrödinger equation with a white-noise potential: Phase-space approach to spread and singularity

2005 ◽  
Vol 212 (3-4) ◽  
pp. 195-204 ◽  
Author(s):  
Albert C. Fannjiang
Keyword(s):  
Phase Space ◽  
Schrödinger Equation ◽  
White Noise ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Space Approach

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).

MEASURE SYNCHRONIZATION IN A COUPLED HAMILTONIAN SYSTEM ASSOCIATED WITH NONLINEAR SCHRÖDINGER EQUATION

2005 ◽  
Vol 19 (15) ◽  
pp. 737-742 ◽  
Author(s):  
U. E. VINCENT ◽  
A. N. NJAH ◽  
O. AKINLADE
Keyword(s):  
Phase Space ◽  
Hamiltonian System ◽  
Invariant Measure ◽  
Schrödinger Equation ◽  
Hamiltonian Systems ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Measure Synchronization

We present preliminary numerical findings concerning measure synchronization in a pair of coupled Nonlinear Hamiltonian Systems (NLHS) derived from a Nonlinear Schrödinger Equation (NLSE). The dynamics of the two coupled NLHS were found to exhibit a transition to coherent invariant measure; their orbits sharing the same phase space as the coupling strength is increased. Transitions from quasiperiodicity (QP) measure desynchronization to QP measure synchronization and QP measure desynchronization to chaotic (CH) measure synchronization were observed.

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