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Smooth and Non-Smooth Traveling Wave Solutions of Some Generalized Camassa-Holm Equations
Title
Smooth and Non-Smooth Traveling Wave Solutions of Some Generalized Camassa-Holm Equations.
Creator
Rehman, Taslima, Choudhury, Sudipto, Nevai, Andrew, Rollins, David, University of Central Florida
Abstract / Description
In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling...
Show moreIn this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, non-smooth M-wave solutions. While the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes.In the second part of the thesis, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbits of their traveling-wave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. Of course, the convergence rate is not comparable to typical asymptotic series. However, asymptotic solutions for global behavior along a full homoclinic/heteroclinic orbit are currently not available.
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Date Issued
2013
Identifier
CFE0004918, ucf:49637
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004918
Nonlinear dispersive partial differential equations of physical relevance with applications to vortex dynamics
Title
Nonlinear dispersive partial differential equations of physical relevance with applications to vortex dynamics.
Creator
VanGorder, Robert, Kaup, David, Vajravelu, Kuppalapalle, Nevai, Andrew, Mohapatra, Ram, Kassab, Alain, University of Central Florida
Abstract / Description
Nonlinear dispersive partial differential equations occur in a variety of areas within mathematical physics and engineering. We study several classes of such equations, including scalar complex partial differential equations, vector partial differential equations, and finally non-local integro-differential equations. For physically interesting families of these equations, we demonstrate the existence (and, when possible, stability) of specific solutions which are relevant for applications....
Show moreNonlinear dispersive partial differential equations occur in a variety of areas within mathematical physics and engineering. We study several classes of such equations, including scalar complex partial differential equations, vector partial differential equations, and finally non-local integro-differential equations. For physically interesting families of these equations, we demonstrate the existence (and, when possible, stability) of specific solutions which are relevant for applications. While multiple application areas are considered, the primary application that runs through the work would be the nonlinear dynamics of vortex filaments under a variety of physical models. For instance, we are able to determine the structure and time evolution of several physical solutions, including the planar, helical, self-similar and soliton vortex filament solutions in a quantum fluid. Properties of such solutions are determined analytically and numerically through a variety of approaches. Starting with complex scalar equations (often useful for studying two-dimensional motion), we progress through more complicated models involving vector partial differential equations and non-local equations (which permit motion in three dimensions). In many of the examples considered, the qualitative analytical results are used to verify behaviors previously observed only numerically or experimentally.
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Date Issued
2014
Identifier
CFE0005272, ucf:50545
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005272
INTEGRAL PROJECTION MODELS REVEAL INTERACTIVE EFFECTS OF BIOTIC FACTORS AND DISTURBANCE ON PLANT DEMOGRAPHY
Title
INTEGRAL PROJECTION MODELS REVEAL INTERACTIVE EFFECTS OF BIOTIC FACTORS AND DISTURBANCE ON PLANT DEMOGRAPHY.
Creator
Tye, Matthew, Quintana-Ascencio, Pedro, Fauth, John, Nevai, Andrew, Menges, Eric, University of Central Florida
Abstract / Description
Understanding factors limiting population growth is crucial to evaluating species persistence in changing environments. I used Integral Projection Models (IPMs) to elucidate the role of biotic interactions and disturbance on population growth rate in two plants: Helianthemum squamatum, a perennial endemic to gypsum habitats in central Spain, and Liatris ohlingerae, a long-lived perennial endemic to the Lake Wales Ridge of central Florida. In H. squamatum, there was a strong positive effect of...
Show moreUnderstanding factors limiting population growth is crucial to evaluating species persistence in changing environments. I used Integral Projection Models (IPMs) to elucidate the role of biotic interactions and disturbance on population growth rate in two plants: Helianthemum squamatum, a perennial endemic to gypsum habitats in central Spain, and Liatris ohlingerae, a long-lived perennial endemic to the Lake Wales Ridge of central Florida. In H. squamatum, there was a strong positive effect of trampling in the site with the highest plant density and moderate positive effects of seed addition in the site with the lowest plant density. Differences in treatment effectiveness between sites may represent a shift from seed to microsite limitation at increasing densities. Additionally, a distinct drop in population growth rate occurred in the hottest and driest year (2009-10). In Liatris ohlingerae, roadside populations had consistently higher population growth rates than scrub populations. A modest negative effect of time-since-fire was observed in plants that did not experience herbivory. Both habitat and time-since-fire showed distinct interactions with vertebrate herbivory, with herbivory increasing the difference in growth rate between habitats and decreasing the difference between time-since-fire classes. The direct effect of herbivory was negative in all environmental combinations except in long unburned populations. These results demonstrate the importance of considering environmental interactions when constructing population models, as well as the validity of using IPMs to assess interactions in species with differing life histories.
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Date Issued
2014
Identifier
CFE0005271, ucf:50558
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005271