M.Sc Student | Gal Fleishon |
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Subject | Long Time Behavior in Non-Linear Dynamics |

Department | Department of Physics |

Supervisor | Professor Emeritus Fishman Shmuel (Deceased) |

Full Thesis text |

In this work Anderson Localization was studied for the Nonlinear Schrödinger Equation (NLSE) in a random potential in the framework of perturbation theory, on a one dimensional lattice. It is a paradigm for competition between strong (exponential)

localization and spreading. If non-linearity is not present, at long time the wave packet is localized due to Anderson Localization while when a random potential is absent spreading occurs. Finding the long time asymptotic behavior of this model is a fundamental problem since heuristic and numerical arguments contradict each other. A renormalized perturbation theory in the nonlinearity parameter that was developed by Krivolapov, Fishman, and Soffer (2009) can be applied to this model for a small non-linearity parameter. In this method the wave function is expanded in powers of the non-linearity parameter. This expansion was conjectured to have an optimal order, that beyond this order, the contribution of each term in the asymptotic series to wave function is large. It was found that the numerical solutions show spreading. It is not clear whether it is a physical phenomenon or a numerical artifact. It is not known which time can be called asymptotic, representing the infinite time behavior. Moreover numerical solutions use the split step method resulting in the solution of a modified model which may cause a deviation from the solution of the original model. In order to compare the results of perturbation theory to the ones of numerical calculations, perturbation theory has to be extended to high orders. In previous work it was performed to fourth order. In the present work it was extended to the sixth order, indicating that for the nonlinearity used, the fifth order is the optimal one for the relevant asymptotic expansion.

This required to develop a sophisticated symbolic algorithm to order all terms in perturbation theory and sum them.