Behavior of solutions to the 1D focusing stochastic nonlinear Schrödinger equation with spatially correlated noise (original) (raw)
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IMA Journal of Applied Mathematics
We study the focusing stochastic nonlinear Schrödinger equation in 1D in the L2L^2L2-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 L2L^2L2-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...
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Journal of Mathematical Analysis and Applications, 2012
We study small random perturbations by additive white-noise of a spatial discretization of a reaction-diffusion equation with a stable equilibrium and solutions that blow up in finite time. We prove that the perturbed system blows up with total probability and establish its order of magnitude and asymptotic distribution. For initial data in the domain of explosion we prove that the explosion time converges to the deterministic one while for initial data in the domain of attraction of the stable equilibrium we show that the system exhibits metastable behavior.
Stochastic nonlinear Schrödinger equations: No blow-up in the non-conservative case
Journal of Differential Equations, 2017
This paper is devoted to the study of noise effects on blow-up solutions to stochastic nonlinear Schrödinger equations. It is a continuation of our recent work [2], where the (local) well-posedness is established in H 1 , also in the non-conservative critical case. Here we prove that in the non-conservative focusing mass-(super)critical case, by adding a large multiplicative Gaussian noise, with high probability one can prevent the blow-up on any given bounded time interval [0, T ], 0 < T < ∞. Moreover, in the case of spatially independent noise, the explosion even can be prevented with high probability on the whole time interval [0, ∞). The noise effects obtained here are completely different from those in the conservative case studied in [5].
Mathematical Models and Methods in Applied Sciences, 2009
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Models of thermal explosion in a closed system are studied at the macroscopic level, where a nonlinear rate equation is solved numerically, at the stochastic level, where the corresponding master equation is solved numericalIy and also analyzed through a 1IN expansion (N is the number of particles in the system) and at the atomistic level, where molecular dynamics simulations of reacting hard disks are carried out. We find that for 800 particles (N= 800) simulation gives sufficient agreement with the macroscopic description of the average concentration. In the region of N= 50-2000 the stochastic and molecular dynamics results show significant overlap with each other; as expected, the effects of fluctuations decrease with increasing N. Under a low-temperature condition (slow reaction rate), a regime which cannot be realized in molecular dynamics simulation, direct numerical solution of the master equation reveals a bimodal distribution during times comparable to a correlation time. This behavior of transient bifurcation, which had been discussed previously, is shown to be a result of small system size.
Metastability for small random perturbations of a PDE with blow-up
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Stochastic Methods and Dynamical Wave-function Collapse
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Physical Review Letters, 2000
We introduce a stochastic partial differential equation capable of reproducing the main features of spatiotemporal intermittency (STI). The model also displays a noise-induced transition from STI to laminarity as the noise intensity increases. Simulations and mean-field analysis are used to characterize this transition and to give a quantitative prescription of its loci in the relevant parameter space. PACS numbers: 05.45.Jn, 05.40.Ca, 05.45.Pq, 47.27.Cn Extended nonlinear dynamical systems often display spatiotemporal chaos (STC) . This is a complex temporally chaotic and spatially incoherent evolution with correlations decaying in both time and space. In spite of considerable theoretical and experimental efforts devoted over the last decade to precisely define STC and its different regimes, the present status is still unsatisfactory. A useful pathway to improve our understanding of STC is the investigation of scenarios based on simple models, with few controlled ingredients, that reproduce its main characteristics. Models of this kind are instrumental in searching for generic mechanisms and universal behavior. However, it is noticeable that very few of the currently available STC scenarios belong to the framework of stochastic partial differential equations (SPDE) . The mapping of the Kuramoto-Shivashinsky equation, paradigm of the STC regime named phase turbulence, to the Kardar-Parisi-Zhang [10] stochastic model for surface growth, is a remarkably successful exceptional example.
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Applied Mathematics and Computation, 2003
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On the stochastic nonlinear Schrodinger equation
HAL (Le Centre pour la Communication Scientifique Directe), 2010
This thesis is devoted to the study of stochastic nonlinear Schrödinger equations (abbreviated as SNLS) with linear multiplicative noise in two aspects: the wellposedness in L 2 (R d), H 1 (R d) and the noise effects on blowup phenomena in the non-conservative case. vii First of all, I would like to use this opportunity to express my sincere gratitude to my supervisors Prof. Dr. Michael Röckner and Prof. Dr. Zhi-Ming Ma. They introduce me into the mathematical world and teach me by personal examples as well as verbal instructions. I benefit so much from their profound knowledge, grand view and illuminating conversations. I am also indebted to them for the great patience and continuous encouragement along the research road. Without their helps this thesis would never have been possible. I am also truly grateful to Prof. Dr. Viorel Barbu for fruitful discussions and inspirations during the research. I am deeply impressed by his enthusiasm, and I am also very thankful for his generous helps in the preparation of this thesis.