Abstract
We study the dynamics of a stochastic nonlinear Schrodinger equation with both a quadratic potential and an additive noise. We show that in both cases of repulsive potential and attractive one, any initial data with finite variance gives birth to a solution that blows up in arbitrarily small time. This is in contrast to the deterministic case when the potential is repulsive, where strong potentials could prevent the solutions from blowing up. Our result hence indicates that the additive noise rather than the potential dominates the dynamical behaviors of the solutions to the stochastic nonlinear Schrodinger equations.
Highlights
On the base of the local wellposedness theory established in [ ], they subsequently investigated the effect of an additive noise on the finite time blow-up behavior of the solutions for a focusing nonlinear Schrödinger equation in [ ]
In this paper, we are interested in the blow-up problem for the following stochastic nonlinear Schrödinger equation with a quadratic potential iut + u + |u| σ u + θ |x| u – η =, t ≥, x ∈ Rn, ( . ) u(, x) = u (x).Here θ ∈ R, σ >, and ηis a complex-valued noise that is white in time and correlated in space
Theorem . says that, for the stochastic Schrödinger equation ( . ) the white noise rather than the potential determines the dynamical behaviors of the solution
Summary
On the base of the local wellposedness theory established in [ ], they subsequently investigated the effect of an additive noise on the finite time blow-up behavior of the solutions for a focusing nonlinear Schrödinger equation in [ ]. As far as we know, the blow-up problem for the Schrödinger equations with a potential and a noise was first studied in [ ], where Fang et al showed that for the initial data with sufficiently negative energy, which is similar to the requirement of deterministic equation, the corresponding solutions blow up in finite time with positive probability.
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