Effective noise theory for the Nonlinear Schrödinger Equation with disorder (original) (raw)
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Effective noise theory for the Nonlinear Schr
2011
For the Nonlinear Shr\"odinger Equation with disorder it was found numerically that in some regime of the parameters Anderson localization is destroyed and subdiffusion takes place for a long time interval. It was argued that the nonlinear term acts as random noise. In the present work the properties of this effective noise are studied numerically. Some assumptions made in earlier work were verified, the dependence of various quantities on the localization length of the linear problem were computed. A scenario for the possible breakdown of the theory for a very long time is outlined.
Subdiffusion in classical and quantum nonlinear Schrödinger equations with disorder
Computers & Mathematics with Applications, 2016
The review is concerned with the nonlinear Schrödinger equation (NLSE) in the presence of disorder. Disorder leads to localization in the form of the localized Anderson modes (AM), while nonlinearity is responsible for the interaction between the AMs and transport. The dynamics of an initially localized wave packets are concerned in both classical and quantum cases. In both cases, there is a subdiffusive spreading, which is explained in the framework of a continuous time random walk (CTRW), and it is shown that subdiffusion is due to the transitions between those AMs, which are strongly overlapped. This overlapping being a common feature of both classical and quantum dynamics, leads to the clustering with an effective trapping of the wave packet inside each cluster. Therefore, the dynamics of the wave packet corresponds to the CTRW, where the basic mechanism of subdiffusion is an entrapping of the wave packet with delay, or waiting, times distributed by the power law w(t) ∼ 1/t 1+α , where α is the transport exponent. It is obtained that α = 1/3 for the classical NLSE and α = 1/2 for the quantum NLSE.
Subdiffusion in the nonlinear Schrödinger equation with disorder
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The nonlinear Schroedinger equation in the presence of disorder is considered. The dynamics of an initially localized wave packet is studied. A subdiffusive spreading of the wave packet is explained in the framework of a continuous time random walk. A probabilistic description of subdiffusion is suggested and a transport exponent of subdiffusion is obtained to be 2/5.
Subdiffusion in the nonlinear Schr�dinger equation with disorder
Phys Rev E, 2010
The nonlinear Schroedinger equation in the presence of disorder is considered. The dynamics of an initially localized wave packet is studied. A subdiffusive spreading of the wave packet is explained in the framework of a continuous time random walk. A probabilistic description of subdiffusion is suggested and a transport exponent of subdiffusion is obtained to be 2/5.
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In this paper we consider the problem of solitary wave propagation in a weakly disordered potential. Through a series of careful numerical experiments we have observed behavior which is in agreement with the theoretical predictions of Kivshar et al., Bronski, and Gamier. In particular we observe numerically the existence of two regimes of propagation. In the first regime the mass of the solitary wave decays exponentially, while the velocity of the solitary wave approaches a constant. This exponential decay is what one would expect from known results in the theory of localization for the linear Schrödinger equation. In the second regime, where nonlinear effects dominate, we observe the anomalous behavior which was originally predicted by Kivshar et al. In this regime the mass of the solitary wave approaches a constant, while the velocity of the solitary wave displays an anomalously slow decay. For sufficiently small velocities (when the theory is no longer valid) we observe phenomena...
EPL (Europhysics Letters), 2012
Localization-delocalization transition in a discrete Anderson nonlinear Schrödinger equation with disorder is shown to be a critical phenomenon -similar to a percolation transition on a disordered lattice, with the nonlinearity parameter thought as the control parameter. In vicinity of the critical point the spreading of the wave field is subdiffusive in the limit t → +∞. The second moment grows with time as a power law ∝ t α , with α exactly 1/3. This critical spreading finds its significance in association with the general problem of transport along separatrices of dynamical systems with many degrees of freedom and is mathematically related with a description in terms fractional derivative equations. Above the delocalization point, with the criticality effects stepping aside, we find that the transport is subdiffusive with α = 2/5 consistently with the results from previous investigations. A threshold for unlimited spreading is calculated exactly by mapping the transport problem on a Cayley tree.
Dynamics of wave packets for the nonlinear Schroedinger equation with a random potential
2009
The dynamics of an initially localized Anderson mode is studied in the framework of the nonlinear Schroedinger equation in the presence of disorder. It is shown that the dynamics can be described in the framework of the Liouville operator. An analytical expression for a wave function of the initial time dynamics is found by a perturbation approach. As follows from a perturbative solution the initially localized wave function remains localized. At asymptotically large times the dynamics can be described qualitatively in the framework of a phenomenological probabilistic approach by means of a probability distribution function. It is shown that the probability distribution function may be governed by the fractional Fokker-Planck equation and corresponds to subdiffusion.
Stochastics and Partial Differential Equations: Analysis and Computations
We study the focusing stochastic nonlinear Schrödinger equation in one spatial dimension with multiplicative noise, driven by a Wiener process white in time and colored in space, in the L 2-critical and supercritical cases. The mass (L 2-norm) is conserved due to the multiplicative noise defined via the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first investigate how the energy is affected by various spatially correlated random perturbations. We then study the influence of the noise on the global dynamics measuring the probability of blow-up versus scattering behavior depending on various parameters of correlation kernels. Finally, we study the effect of the spatially correlated noise on the blow-up behavior, and conclude that such random perturbations do not influence the blow-up dynamics, except for shifting of the blow-up center location. This is similar to what we observed in [32] for a space-time white driving noise. Contents 1. Introduction 2. Preliminaries 2.1. Time dependence of mass and energy 2.2. Discretizations and numerical schemes 3. Stochastic perturbation driven by a Q-Wiener process 3.1. Description of the driving noise 3.2. Discrete mass and energy; upper bounds on energy 3.3. Numerical tracking of discrete energy 4. Stochastic perturbation driven by a homogeneous Wiener process 4.1. Description of the driving noise 4.2. Covariance matrix computation, bounds on discrete energy 4.3. Numerical tracking of discrete energy 5. Influence of noise on global behavior: blow-up probability 5.1. Comments about mesh-refinement 5.2. Probability of blow-up 6. Effect of the noise on blow-up dynamics 7. Conclusion Appendix A Appendix B References