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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
The following study will address the questions: "Why is it essential to teach elementary students about sustainability?" and "How do we teach elementary students about sustainability?" Teachers have an obligation to their students, as well as to the planet, to make their students concerned about sustainability. Many natural resources students need in the future in order to survive are running out. Without a future generation prepared with the skills to challenge issues and investigate complex problems, the Earth will remain in a jeopardized state. Teachers need to incorporate sustainability-themed literature into their classrooms and lessons in order to prepare this future generation with those skills. Teachers should inform their students about the history of the term "sustainability." During this study, it was found that the sustainability curriculum topics and the "Four Ways of Thinking" could have been included into the existing curriculum. Subsequently, sustainable and critical thinking are aligned because they both share many of the same skills. Teachers could have students investigate current and past news articles to discover the problems caused by using natural resources unsustainably. Current news articles could be given to students, so they can look at how these issues can be solved with the use of alternative resources. Many of the younger students might not have a high enough reading level to understand news articles. There have been websites created that are geared toward younger audiences, so this would allow teachers to incorporate news into their lessons. Projects and class discussions should be rooted in sustainability. Class discussions can take place every day or once a week, while projects can occur over the course of a single month. Many teachers think the curriculum is too focused on improving state test scores. Nevertheless, the curriculum should contain sustainable and critical thinking skills. The implementation of sustainability education seems to overwhelm teachers because some do not see how they can incorporate it into their classrooms. However, this study found that these particular instructors can design existing lesson topics around the content and ways of thinking in sustainability education. Another reason why there is resistance to sustainability education is because the sustainability programs would add even more to each school's budget. Schools could raise funds for sustainability education, or apply for grants from the government. The in-depth literature review within this qualitative and open-ended study looked at subjective data. Sustainability, sustainability education, elementary curriculum, classroom, and teachers were just a sample of the key terms used for article searches in Google Scholar through Arizona State University. The reduction techniques included discarding any literature that neither linked directly to the problem statement nor with the ideas relating to the research questions. Limitations within the field of sustainability and elementary education include the research among middle and high schools across the nation. Many of the ideas for future research include the analysis of the long-term effects of incorporation of sustainability education within elementary curricula.
ContributorsWenninger, Jessica Shea (Author) / Ahmad, Omaya (Thesis director) / Shelton, Catharyn (Committee member) / Division of Teacher Preparation (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05