The existence and stability of solitons in discrete nonlinear Schrödinger equations

Syafwan, Mahdhivan (2012) The existence and stability of solitons in discrete nonlinear Schrödinger equations. PhD thesis, University of Nottingham.

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Abstract

In this thesis, we investigate analytically and numerically the existence and stability of discrete solitons governed by discrete nonlinear Schrödinger (DNLS) equations with two types of nonlinearity, i.e., cubic and saturable nonlinearities. In the cubic-type model we consider stationary discrete solitons under the effect of parametric driving and combined parametric driving and damping, while in the saturable-type model we examine travelling lattice solitons.

First, we study fundamental bright and dark discrete solitons in the driven cubic DNLS equation. Analytical calculations of the solitons and their stability are carried out for small coupling constant through a perturbation expansion. We observe that the driving can not only destabilise onsite bright and dark solitons, but also stabilise intersite bright and dark solitons. In addition, we also discuss a particular application of our DNLS model in describing microdevices and nanodevices with integrated electrical and mechanical functionality.

By following the idea of the work above, we then consider the cubic DNLS equation with the inclusion of parametric driving and damping. We show that this model admits a number of types of onsite and intersite bright discrete solitons of which some experience saddle-node and pitchfork bifurcations. Most interestingly, we also observe that some solutions undergo Hopf bifurcations from which periodic solitons (limit cycles) emerge. By using the numerical continuation software Matcont, we perform the continuation of the limit cycles and determine the stability of the periodic solitons.

Finally, we investigate travelling discrete solitons in the saturable DNLS equation. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton-Raphson method to find traveling solitons with non-oscillatory tails, i.e., embedded solitons. A variational approximation (VA) is also applied to examine analytically the travelling solitons and their stability, as well as to predict the location of the embedded solitons.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Susanto, H.
Cox, S.M.
Subjects: Q Science > QC Physics > QC170 Atomic physics. Constitution and properties of matter
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 12916
Depositing User: EP, Services
Date Deposited: 06 Sep 2013 08:48
Last Modified: 13 Sep 2016 17:05
URI: http://eprints.nottingham.ac.uk/id/eprint/12916

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