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Title: Finite-gap solutions of the defocusing nonlinear Schrodinger equation
Author: Baek, Chin-wook
ISNI:       0000 0001 3435 2184
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2008
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The subject of this thesis is a study of the two-component nonlinear Schrodinger equation (NSE) in (1+1)-dimension. Two aitemative cases are possible, referred to as the focusing and defocusing cases in applications to studies in nonlinear optics. The main focus of the work in this thesis is on the defocusing case, but results known for the focusing case are also discussed and compared, where relevant. This thesis is comprised of two themes. The first is the effective integration of the NSE, using techniques of algebrogeometric theory. Here, the Baker-Akheizer formalism is developed and applied to the Manakov system. This formalism is used to derive the finite-gap solution, expressed in terms of Riemann theta-functions. The second is the derivation of soliton solutions from these finite-gap solutions, by considering an important limit; namely, the closing ofthe gap in the Riemanu surface associated with. the spectral curve yields explicit representations for soliton solutions. The scalar NSE is first considered by way of introducing and discussing relevant techniques. Then, it is shown how these methods can be extended to the vector case. Next, two particular cases are considered. Firstly, it is shown that the genus 1 case yields a dark-dark soliton solution in the limiting case. An aspect of the vector problem not found with the scalar case is the existence of dark-bright solitons. A single dark-bright soliton can be obtained by considering genus 2, not genus 1, in the soliton limit. A discussion of these derivations is an important feature of the thesis. We discuss next a solution obtained elsewhere introducing a separable ansatz for both components shown to be a genus 2 solution. We discuss the fact that the appropriate curve is hyperelliptic, and indicate how it is related to our trigonal curve by a birational map.
Supervisor: Not available Sponsor: Not available
Qualification Name: Imperial College, 2008 Qualification Level: Doctoral
EThOS ID:  DOI: Not available