Thesis-1997-Zakynthinaki.pdf (1.34 MB)
Numerical solution of the non-linear Schroedinger equation : the half-line problem and dynamical systems and bifurcations of vector fields
thesis
posted on 2014-02-17, 15:08 authored by Maria S. ZakynthinakiSolutions to the nonlinear Schrodinger equation with potential V(u) =
-λulul2 have been theoretically and numerically calculated, revealing the
formation of solitons. In this study the finite element method with linear
basis functions, distinguished for its simplicity and effective applicability,
is considered and a predictor-corrector scheme is applied to simulate the
propagation in time. Numerical experiments include the propagation of a
single soliton form, a two-soliton collision, as well as the formation of more
than one solitons from non-soliton initial data. The important problem of
boundary reflections has been successfully overcome by the implementation
of absorbing boundaries, a method that in practice achieves a gradual reduction
of the wave amplitude at the end of each time step.
The second part of this work deals with dynamical systems of the form [see file]. The dynamics of such systems near their equilibrium
point depends strongly on the adjustable parameter μ, as it is possible for the
system to lose its hyperbolicity and a bifurcation to occur. After reviewing
aspects of linearisation, the prospect of change in the equilibrium solutions
has been studied, both for flows and maps, in terms of the eigenvalues of
the linearised system. In the study of steady-state bifurcation, elements of
saddle-node, transcritical, pitchfork, as well as period-doubling bifurcation
are considered. Finally, the case when equilibrium solutions persist, known
as Hopf bifurcation, has also been included.
History
School
- Science
Department
- Mathematical Sciences
Publisher
© Maria S. ZakynthinakiPublication date
1997Notes
A Master's Thesis. Submitted in partial fulfilment of the requirements for the award of Master of Philosophy of Loughborough University.Language
- en