scholarly journals FUNCTIONAL-INTEGRAL SOLUTION FOR THE SCHRÖDINGER EQUATION WITH POLYNOMIAL POTENTIAL: A WHITE NOISE APPROACH

2011 ◽  
Vol 14 (04) ◽  
pp. 675-688 ◽  
Author(s):  
SONIA MAZZUCCHI
Keyword(s):  
Weak Solution ◽  
Schrödinger Equation ◽  
Integral Representation ◽  
White Noise ◽  
Schrodinger Equation ◽  
Functional Integral ◽  
Integral Solution ◽  
Polynomial Potential ◽  
White Noise Functional ◽  
Functional Integral Representation

A functional integral representation for the weak solution of the Schrödinger equation with polynomially growing potentials is proposed in terms of a white noise functional.

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.

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