Bifurcations in nonlinear Schrödinger equations with double well potentials
Kim, Hee Yeon
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https://hdl.handle.net/2142/104763
Description
Title
Bifurcations in nonlinear Schrödinger equations with double well potentials
Author(s)
Kim, Hee Yeon
Issue Date
2019-04-02
Director of Research (if dissertation) or Advisor (if thesis)
Kirr, Eduard Wilhelm
Doctoral Committee Chair(s)
Laugesen, Richard S.
Committee Member(s)
Bronski, Jared C.
Hur, Vera Mikyoung
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2019-08-23T19:51:33Z
Keyword(s)
Schrödinger equation
Bifurcation
Bound states
Double wells
Abstract
In this thesis, we consider nonlinear Schrödinger equations with double well potentials with attractive and repelling nonlinearities. We discuss bifurcations along bound states, especially ground states and the first excited states, and also deal with orbital stability of the ground states. In attractive case with large separations for double wells, our results shows that the ground state must undergo the secondary symmetry breaking bifurcation, while the first excited states can be uniquely extended as long as the bifurcation of the ground state has not occurred. In repelling case with large separations for double wells, we prove that the secondary bifurcation of the ground state does not emerge, even in the strongly nonlinear regime, while the first excited state must undergo the secondary bifurcation on the first excited states.
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