Ph.D Student | Bar-Lev Yevgeny |
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Subject | Anderson Localization in Some Nonlinear Systems |

Department | Department of Physics |

Supervisor | Professor Emeritus Shmuel Fishman (Deceased) |

Full Thesis text |

In this work a perturbation theory for the Nonlinear Schrödinger Equation (NLSE) in one dimension and in the presence of a random potential was developed. This equation was motivated by classical nonlinear wave optics in disordered materials and by studies of Bose-Einstein Condensates (BEC) in disordered optical potentials. Here the NLSE is used to explore the competition between effects of disorder and nonlinearity. In the absence of nonlinearity the NLSE reduces to the Anderson model where in one dimension all the eigenstates are exponentially localized. Consequently, a wavepacket will remain localized near its initial position. In the absence of the random potential spreading takes place. Various contradicting arguments concerning the asymptotic behavior in space and time were put forward. Nevertheless, in spite of the fact that many numerical calculations were performed their significance is not clear since it is not known what is the time scale that determines when long time asymptotics is achieved. Therefore an analytic approach is required.

In this work we have developed a perturbation theory in the nonlinearity parameter. This approach enabled us to show that there is a front which propagates not faster than logarithmically in time. Beyond this front the wavepacket remains exponentially bounded in space, as in the case for the linear disordered system. The perturbation expansion also allowed us to obtain a numerical solution to the NLSE with some control of the error. The development of the perturbation theory required to overcome many difficulties which are typical to nonlinear differential equations and it is therefore relevant for the study of other nonlinear problems, e.g. the Fermi-Pasta-Ulam (FPU) problem.