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Description
Title: | Asymptotic Stability of the Ground States of the Nonlinear Schrodinger Equation |
Author(s): | Mizrak, Ozgur |
Doctoral Committee Chair(s): | Jared Bronski |
Department / Program: | Mathematics |
Discipline: | Mathematics |
Degree Granting Institution: | University of Illinois at Urbana-Champaign |
Degree: | Ph.D. |
Genre: | Dissertation |
Subject(s): | Mathematics |
Abstract: | We consider a class of nonlinear Schrodinger equations in N = 3, 4, 5 space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L2( RN )) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that "shadows" the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations. |
Issue Date: | 2009 |
Type: | Text |
Language: | English |
Description: | 91 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2009. |
URI: | http://hdl.handle.net/2142/86924 |
Other Identifier(s): | (MiAaPQ)AAI3363042 |
Date Available in IDEALS: | 2015-09-28 |
Date Deposited: | 2009 |
This item appears in the following Collection(s)
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Dissertations and Theses - Mathematics
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Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois