M.Sc Student | Michaely Erez |
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Subject | Effective Noise Theory for the Nonlinear Schrodinger Equation |

Department | Department of Physics |

Supervisor | Professor Emeritus Shmuel Fishman (Deceased) |

Full Thesis text |

The Nonlinear Schrödinger Equation (NLSE) with a random potential is a paradigm for competition between randomness and nonlinearity. It is also of experimental relevance for experiments in optics and in atom optics. In spite of extensive exploration for the last two decades the elementary properties of its dynamics are still not understood. Dynamical localization is found for the linear Anderson model. In particular in one dimension in the presence of a random potential typically a wave packet that is initially localized will remain localized for arbitrary long time. The elementary open question is whether this holds also in the presence of nonlinearity. In numerical calculations it is found that spreading by subdiffusion takes place for a wide range of parameters and for a long time. Analytical and rigorous arguments indicate that this subdiffusion cannot be asymptotic in time. The basic justification that was given for the subdiffusion is that the nonlinear term acts as noise. In this thesis this assumption was reformulated and tested numerically. It was found that in the relevant regime the nonlinear term behaves as noise with a rapidly decaying correlation function, required for subdiffusion. It was also found that this effective noise is stationary. A scenario for the failure of the effective theory in the long time limit is outlined. Several statistical properties of the linear Anderson model that are relevant for the NLSE with a random potential were calculated. A “toy model” for the NLSE was proposed.