Kusdiantara, Rudy (2018) Homoclinic snaking in discrete systems. PhD thesis, University of Essex.

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Abstract
In this thesis, we investigate analytically and numerically bifurcations of localized solutions in discrete systems, i.e., the discrete SwiftHohenberg, an optical cavity equation, and the discrete AllenCahn equation, which have infinitely multiplicity called homoclinic snaking. First, we study the discrete SwiftHohenberg equation with cubic and quintic nonlinearity, obtained from discretizing the spatial derivatives of the SwiftHohenberg equation using central finite differences. We investigate the discretization effect on the bifurcation behavior, where we identify three regions of the coupling parameter, i.e., strong, weak, and intermediate coupling. In the intermediate coupling region, multiple Maxwell points can occur for the periodic solutions and may cause irregular snaking and isolas. Theoretical analysis for the snaking and stability of the corresponding solutions is provided in the weak coupling region. Next, we study timeindependent solutions of an optical cavity equation with saturable nonlinearity. When the nonlinearity is of Kerrtype (i.e., cubic), one obtains the discrete version of LugiatoLefever equation. The equation admits uniform and localized solutions. Localized solutions can be formed by combining two different uniform states, which can develop a snaking structure in their bifurcation diagram when a control parameter is varied, i.e., homoclinic snaking. Cshaped isolas may also occur when the background of localized states disappear at a certain bifurcation parameter value. The semianalytical approximation is also proposed to determine the stability of the corresponding solutions. Finally, we present a study on timeindependent solutions of the twodimensional discrete AllenCahn equation with cubic and quintic nonlinearity. Three different types of lattices are considered, i.e., square, honeycomb, and triangular lattices. Localized solutions of discrete AllenCahn equation also can be formed by combining two different uniform states. We introduce an activecell approximation, which is extended from the oneactive site approximation in onedimensional case for a weakly coupled system.
Item Type:  Thesis (PhD) 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science and Health > Mathematical Sciences, Department of 
Depositing User:  Rudy Kusdiantara 
Date Deposited:  06 Nov 2018 15:15 
Last Modified:  06 Nov 2018 15:15 
URI:  http://repository.essex.ac.uk/id/eprint/23314 
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