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Description
Nonlinear dispersive equations model nonlinear waves in a wide range of physical and mathematics contexts. They reinforce or dissipate effects of linear dispersion and nonlinear interactions, and thus, may be of a focusing or defocusing nature. The nonlinear Schrödinger equation or NLS is an example of such equations. It appears as a model in hydrodynamics, nonlinear optics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. In mathematics, one of the interests is to look at the wave interaction: waves propagation with different speeds and/or different directions produces either small perturbations comparable with linear behavior, or creates solitary waves, or even leads to singular solutions. This dissertation studies the global behavior of finite energy solutions to the $d$-dimensional focusing NLS equation, $i partial _t u+Delta u+ |u|^{p-1}u=0, $ with initial data $u_0in H^1,; x in Rn$; the nonlinearity power $p$ and the dimension $d$ are chosen so that the scaling index $s=frac{d}{2}-frac{2}{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $ME[u_0]<1$ ($ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient $g_u$ of a solution $u$ to NLS is initially less than 1, i.e., $g_u(0)<1,$ then the solution exists globally in time and scatters in $H^1$ (approaches some linear Schr"odinger evolution as $ttopminfty$); if the renormalized gradient $g_u(0)>1,$ then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of $H^1$ norm in infinite time. This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle. One of the difficulties is fractional powers of nonlinearities which are overcome by considering Besov-Strichartz estimates and various fractional differentiation rules.
ContributorsGuevara, Cristi Darley (Author) / Roudenko, Svetlana (Thesis advisor) / Castillo_Chavez, Carlos (Committee member) / Jones, Donald (Committee member) / Mahalov, Alex (Committee member) / Suslov, Sergei (Committee member) / Arizona State University (Publisher)
Created2011
Description
Arachnids belong to the phylum Arthropoda, the largest phylum in the animal kingdom. Ticks are blood-feeding arachnids that vector numerous pathogens of significant medical and veterinary importance, while scorpions have become a common concern in urban desert cities due to the high level of toxicity in their venom. To date, viruses associated with arachnids have been under sampled and understudied. Here viral metagenomics was used to explore the diversity of viruses present in ticks and scorpions. American dog ticks (Dermacentor variabilis) and blacklegged ticks (Ixodes scapularis) were collected in Pennsylvania while one hairy scorpion (Hadrurus arizonensis) and four bark scorpions (Centruroides sculpturatus) were collected in Phoenix. Novel viral genomes described here belong to the families Polyomaviridae, Anelloviridae, Genomoviridae, and a newly proposed family, Arthropolviridae.
Polyomaviruses are non-enveloped viruses with a small, circular double-stranded DNA (dsDNA) genomes that have been identified in a variety of mammals, birds and fish and are known to cause various diseases. Arthropolviridae is a proposed family of circular, large tumor antigen encoding dsDNA viruses that have a unidirectional genome organization. Genomoviruses and anelloviruses are ssDNA viruses that have circular genomes ranging in size from 2–2.4 kb and 2.1–3.8 kb, respectively. Genomoviruses are ubiquitous in the environment, having been identified in a wide range of animal, plant and environmental samples, while anelloviruses have been associated with a plethora of animals.
Here, 16 novel viruses are reported that span four viral families. Eight novel polyomaviruses were recovered from bark scorpions, three arthropolviruses were recovered from dog ticks and one arthropolvirus from a hairy scorpion. Viruses belonging to the families Polyomaviridae and Arthropolviridae are highly divergent. This is the first more extensive study of these viruses in arachnids. Three genomoviruses were recovered from both dog and deer ticks and one anellovirus was recovered from deer ticks, which are the first records of these viruses being recovered from ticks. This work highlights the diversity of dsDNA and ssDNA viruses in the arachnid population and emphasizes the importance of performing viral surveys on these populations.
Polyomaviruses are non-enveloped viruses with a small, circular double-stranded DNA (dsDNA) genomes that have been identified in a variety of mammals, birds and fish and are known to cause various diseases. Arthropolviridae is a proposed family of circular, large tumor antigen encoding dsDNA viruses that have a unidirectional genome organization. Genomoviruses and anelloviruses are ssDNA viruses that have circular genomes ranging in size from 2–2.4 kb and 2.1–3.8 kb, respectively. Genomoviruses are ubiquitous in the environment, having been identified in a wide range of animal, plant and environmental samples, while anelloviruses have been associated with a plethora of animals.
Here, 16 novel viruses are reported that span four viral families. Eight novel polyomaviruses were recovered from bark scorpions, three arthropolviruses were recovered from dog ticks and one arthropolvirus from a hairy scorpion. Viruses belonging to the families Polyomaviridae and Arthropolviridae are highly divergent. This is the first more extensive study of these viruses in arachnids. Three genomoviruses were recovered from both dog and deer ticks and one anellovirus was recovered from deer ticks, which are the first records of these viruses being recovered from ticks. This work highlights the diversity of dsDNA and ssDNA viruses in the arachnid population and emphasizes the importance of performing viral surveys on these populations.
ContributorsSchmidlin, Kara (Author) / Varsani, Arvind (Thesis advisor) / Van Doorslaer, Koenraad (Committee member) / Stenglein, Mark (Committee member) / Arizona State University (Publisher)
Created2019