Thesis ( Ph.D. ) -- University of Texas at Arlington, 2008.

ABSTRACT The quest to model wave propagation in various physical systems has produced a large set of diverse nonlinear equations. Nonlinear singular integro-differential equations rank amongst the intricate nonlinear wave equations available to study the classical problem of wave propagation in physical systems. Integro-differential equations are characterized by the simultaneous presence of integration and differentiation in a single equation. Substantial interest exists in nonlinear wave equations that are amenable to the Inverse Scattering Transform (IST). The IST is an adroit mathematical technique that delivers analytical solutions of a certain type of nonlinear equation: soliton equation. Initial value problems of numerous physically significant nonlinear equations have now been solved through elegant and novel implementations of the IST. The prototype nonlinear singular integro-differential equation receptive to the IST is the Intermediate Long Wave (ILW) equation, which models one-dimensional weakly nonlinear internal wave propagation in a density stratified fluid of finite total depth. In the deep water limit the ILW equation bifurcates into a physically significant nonlinear singular integro-differential equation known as the 'Benjamin-Ono' (BO) equation; the shallow water limit of the ILW equation is the famous Korteweg-de Vries (KdV) equation. Both the KdV and BO equations have been solved by dissimilar implementations of the IST. The Modified Korteweg-de Vries (MKdV) equation is a nonlinear partial differential equation, which was significant in the historical development of the IST. Solutions of the MKdV equation are mapped by an explicit nonlinear transformation known as the 'Miura transformation' into solutions of the KdV equation. Historically, the Miura transformation manifested the intimate connection between solutions of the KdV equation and the inverse problem for the one-dimensional time independent Schroedinger equation. In light of the MKdV equation's significance, it is natural to seek 'modified' versions of the ILW and BO equations. Solutions of each modified nonlinear singular integro-differential equation should be mapped by an analogue of the original Miura transformation into solutions of the 'unmodified' equation. In parallel with the limiting cases of the ILW equation, the modified version of the ILW equation should reduce to the MKdV equation in the shallow water limit and to the modified version of the BO equation in the deep water limit. The Modified Intermediate Long Wave (MILW) and Modified Benjamin-Ono (MBO) equations are the two nonlinear singular integro-differential equations that display all the required attributes. Several researchers have shown that the MILW and MBO equations exhibit the signature characteristic of soliton equations. Despite the significance of the MILW and MBO equations to soliton theory, and the possible physical applications of the MILW and MBO equations, the initial value problems for these equations have not been solved. In this thesis we use the IST to solve the initial value problems for the MILW and MBO equations on the real-line. The only restrictions that we place on the initial values for the MILW and MBO equations are that they be real-valued, sufficiently smooth and decay to zero as the absolute value of the spatial variable approaches large values.

Thesis (Ph. D.)--University of Sydney, 1999. / Title from title screen (viewed Apr. 21, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliography. Also available in print form.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 85-88). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 85-88). Also available on the Internet.

Thesis (Sc. D.)--Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 1987. / "January 1987." Bibliography: p. 193-200.

We consider the analysis of the scattering problem. Assume that an incoming time harmonic wave is scattered by a surface of an impenetrable obstacle. The reflected wave is determined by the surface impedance of the obstacle. In this paper we will investigate the problem of choosing the surface impedance so that a desired scattering amplitude is achieved. We formulate this control problem within the framework of the minimization of a Tikhonov functional. In particular, questions of the existence of an optimal solution and the derivation of the optimality conditions will be addressed. / Master of Science

For certain initial data, we solve the Novikov-Veselov equation by the inverse scat- tering method. This is a (2+1)-dimensional completely integrable system that gen- eralizes the (1+1)-dimensional Korteweg-de-Vries equation. The method used is the inverse scattering method. To study the direct and inverse scattering maps, we prove existence and uniqueness properties of exponentially growing solutions of the two- dimensional Schrodinger equation. For conductivity-type potentials, this was done by Nachman in his work on the inverse conductivity problem. Our work expands the set of potentials for which the analysis holds, completes the study of the inverse scattering map, and show that the inverse scattering method yields global in time solutions to the Novikov-Veselov equation. This is the first proof that the inverse scattering method yields classical solutions to the Novikov-Veselov equation for the class of potentials considered here.

Optical Coherence Tomography (OCT) is a coherence-gated, micrometer-resolution imaging technique that focuses a broadband near-infrared laser beam to penetrate into optical scattering media, e.g. biological tissues. The OCT resolution is split into two parts, with the axial resolution defined by half the coherence length, and the depth-dependent lateral resolution determined by the beam geometry, which is well described by a Gaussian beam model. The depth dependence of lateral resolution directly results in the defocusing effect outside the confocal region and restricts current OCT probes to small numerical aperture (NA) at the expense of lateral resolution near the focus. Another limitation on OCT development is the presence of a mixture of speckles due to multiple scatterers within the coherence length, and other random noise. Motivated by the above two challenges, a multiple scattering model based on Rytov approximation and Gaussian beam optics is proposed for the OCT setup. Some previous papers have adopted the first Born approximation with the assumption of small perturbation of the incident field in inhomogeneous media. The Rytov method of the same order with smooth phase perturbation assumption benefits from a wider spatial range of validity. A deconvolution method for solving the inverse problem associated with the first Rytov approximation is developed, significantly reducing the defocusing effect through depth and therefore extending the feasible range of NA.

This thesis reports on my Ph.D. research in the area of microwave remote sensing of the Arctic. The main objective of this research is to reconstruct the dielectric profile of the snow-covered sea ice, and indirectly retrieve some of its geophysical and thermodynamic properties. To meet this objective, a nonlinear electromagnetic inverse scattering algorithm is developed that consists of forward and inverse solvers. The input to this algorithm is the normalized radar cross section (NRCS) data collected by radar systems from the snow-covered sea ice profile. The proposed inversion algorithm iteratively minimizes a discrepancy between the measured and simulated NRCS data to achieve an accurate reconstruction. Two main challenges associated with this inverse problem are its ill-posedness and its limited available scattering data. To tackle these, the utilization of appropriate regularization and weighting schemes as well as the incorporation of prior information into the inversion algorithm are employed. These include the utilization of (i) appropriate weighting factors for the misfit cost function, (ii) more sensitive NRCS data with respect to the unknown parameters, (iii) further parametrization of the profile based on the expected distribution, (iv) time-series NRCS data to better initialize the inversion process, and (v) NRCS data collected by the satellite and on-site scatterometer to be inverted simultaneously for profile reconstruction. The experimental data utilized are collected by the author in collaboration with the Centre for Earth Observation Science. These measurements are performed on (i) the artificially-grown sea ice in the Sea-ice Environmental Research Facility, located at the University of Manitoba during winter 2014, and (ii) the landfast sea ice located in the Arctic (Cambridge Bay, Nunavut) during May 2014. The measurement procedure includes NRCS data collection through an on-site C-band scatterometer and a spaceborne SAR satellite and physical sampling of the snow and sea ice. The proposed electromagnetic inverse scattering algorithm is utilized to invert these experimental data sets, as well as some synthetic data sets. It will be shown that the use of various techniques developed in this thesis in conjunction with the developed inversion algorithm results in reasonable snow-covered sea ice profile reconstruction. / October 2016