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Exact topological soliton solutions of the dispersive, strongly perturbed, and 2d sine-gordon type equations

Show simple item record Johnson, Stephen 2018-10-09T05:01:28Z 2018-10-09T05:01:28Z
dc.description.abstract This dissertation uses the Ansatz method to solve for exact topological soliton solutions to sine-Gordon type equations. Single, double, and triple sine-Gordon and sine-cosine-Gordon equations are investigated along with dispersive and highly dispersive variations. After these solutions are found, strong perturbations are added to each equation and the new solutions are found. In solving both the perturbed and unperturbed sine-Gordon type equations, constraints are imposed on the parameters. After finding exact solutions to the dispersive sine-Gordon type equations, three new solutions to the 2D sine-Gordon equation are found. These solutions include the domain wall, the breather, and the domain wall collision. Of particular interest is the Domain wall collision to the 2D sine-Gordon equation, which to the authors' knowledge had not previously been presented in the literature. The first chapter will begin by giving the historical context of solitons and the sine-Gordon equation. It will be shown here that the results found in the later chapters will be important to the study of Josephson junctions, crystal dislocations, ultra-short optical pulses, relativistic field theory, and elementary particles. This chapter will continue on to show the derivation of the discrete sine-Gordon equation by means of the Hamiltonian. The continuous sine-Gordon equation will be shown to arise from the discrete version in the long wave limit. This approximation will also show how the higher order dispersion terms arise. This chapter will conclude by explaining the Ansatz method in detail. It will also show some other common methods for the study of solitons. The second, third, and fourth chapters will find exact solutions to the strongly perturbed sine-Gordon type equations using the Ansatz method. The sine-Gordon type equations studied will include the single, double, and triple sine-Gordon equations, and all of their sine-cosine-Gordon analogs. In addition to these equations, higher order dispersive versions will also be studied. These will include both fourth and sixth order dispersion. The fifth chapter will find exact solutions to the 2D sine-Gordon equation. This study will also be carried out using the Ansatz method. However, in this case special relativity will be used to turn a stationary solution into a moving solution. This will be performed in the analog sense to how it is commonly done for the 1D sine-Gordon equation. The sixth chapter will summarize the dissertation and give some final remarks.
dc.title Exact topological soliton solutions of the dispersive, strongly perturbed, and 2d sine-gordon type equations 2018-09-25T19:03:04Z
dc.language.rfc3066 en

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