University of Illinois Urbana-Champaign

Asymptotic Stability of the Ground States of the Nonlinear Schrodinger Equation

Mizrak, Ozgur

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https://hdl.handle.net/2142/86924

Description

Title
Asymptotic Stability of the Ground States of the Nonlinear Schrodinger Equation
Author(s)
Mizrak, Ozgur
Issue Date
2009
Doctoral Committee Chair(s)
Jared Bronski
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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."
Graduation Semester
2009
Type of Resource
text
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http://hdl.handle.net/2142/86924

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