Numerical Investigations Of The Complex Swift-Hohenberg Equation
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Date
2017-08
Authors
Khairudin, Nur Izzati
Journal Title
Journal ISSN
Volume Title
Publisher
Universiti Sains Malaysia
Abstract
Complex Swift-Hohenberg equation (CSHE) is a model used to study various aspects
of pattern formation in applied mathematics and mathematical physics. Essentially,
the CSHE can be considered as generalization of the nonlinear Schrödinger
equation, an equation that exhibit solitary wave solution. It is known that solitary
wave solutions were used to detect physical phenomena such as the tsunami wave. In
this research, investigation on some of the properties of the CSHE and its solutions
were conducted. This research is divided into three parts. Firstly, the stability of fixed
points of the CSHE using a modified variational formulation was analyzed. In the
second part, investigation on an approximate soliton solution to the CSHE using the
cosine function method was discussed. The final part discussed the stability of the
fixed points for two cases of the dispersionless CSHE, namely for the dispersion-free
and for the destabilized dispersion. The modified variational formulation was reused in
the third part. From the first part, the system exhibit a focus and stationary soliton. Due
to the stability of Jacobian eigenvalues, the fixed points of the CSHE were stable. The
findings from the first part have shown that no bifurcation existed in the transition from
fixed points to limit cycle in the system. The contribution occurred in the second part
where an analytical solution to the CSHE was obtained. The existence of frequency
in the system was established due to the nonlinear gain in the CSHE. The findings
from the second part have shown that the propagation of plain stationary pulses was
obtained in the system while varying the values of nonlinear gain and frequency. In the
third part, the system depended on nonlinearity, since the dispersion essentially has no
effect on the system. It was found that the fixed points of dispersionless CSHE were
stable when the system evaluated at the nonlinear gain and linear loss. Stationary and
exploding solitons have been found under this problem. From the findings, the properties
of the CSHE and its solutions can be extended to understand several other pattern
formations in both cavity solitons and spatiotemporal dissipative solitons.
Description
Keywords
Complex Swift-Hohenberg equation is a model used to study , various aspects of pattern formation in applied mathematics.