In this paper we construct explicit examples of solutions to certain nonlinear wave equations. These semilinear equations are the simplest equations known to possess localized solitary waves in more that one spatial dimension. We construct explicit localized standing wave solutions, which generate multidimensional localized traveling solitary waves under the action of velocity boosts. We study the case of two spatial dimensions and a piecewise-linear nonlinearity. We obtain a large subset of the infinite family of standing waves, and we exhibit several interesting features of the family. Our solutions include solitary waves that carry nonzero angular momenta in their rest frames. The spatial profiles of these solutions also furnish examples of symmetry breaking for nonlinear elliptic equations.

The basic concept to be used in studying the question of one-particle interpretations of relativistic wave equations is that of observables and operator representations that are different from the more usual classically motivated observables and representations. In particular, the concept of a mean-position observable will be used to determine to what extent the one-particle "problems" can be resolved.

In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in [0,π]xR and an existence of a positive solution for a non-positone Dirichlet problem in a bounded subset of R^n.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 135-143). Available also in a digital version from Dissertation Abstracts.

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

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We carry out a preliminary group classification of the following family of non-linear wave equations u_tt =f(u_x)u_xx+g(u_x)+x. We first re-obtain the principal Lie algebra obtained by Ibragimov et al[3) and then construct the equivalence Lie algebra. In order to partially classify this family of wave equations, optimal systems of one-dimensional sub-algebras of the equivalence Lie algebra are constructed and in so doing, two distinct equations are obtained. We furthermore determine some invariant solutions of these equations. / Thesis (MSc. Mathematics) North-West University, Mafikeng Campus, 2003

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Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (leaves 94-98).

Orientadores: Luis Adauto da Justa Medeiros, Cicero Lopes Frota / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T22:02:49Z (GMT). No. of bitstreams: 1 Vicente_Andre_D.pdf: 1601473 bytes, checksum: 8d8834d3fb9f839045ab52e1a9f4887e (MD5) Previous issue date: 2010 / Resumo: Neste trabalho estudamos a resolubilidade de alguns problemas de valores iniciais e de fronteira, envolvendo equações de ondas com Condições de Fronteira da Acústica. Inicialmente, provamos a existência e unicidade de solução para dois problemas com a formulação clássica, introduzida por Beale e Rosencrans [5] (considerando que a fronteira é localmente reagente). Posteriormente provamos resultados para problemas envolvendo uma formulação das condições de fronteira em que consideramos o caso de fronteira não localmente reagente. Para os resultados de existência de solução usamos o método construtivo de Faedo-Galerkin e argumentos de compacidade. Resultados sobre o decaimento de solução baseiam-se no método de Nakao. / Abstract: In this work we study the resolubility of some problems for wave equations with Acoustic Boundary Conditions. Firstly, we prove the existence and uniqueness of solution for two problems with classical formulation due to Beale and Rosencrans [5] (considering locally reacting boundary). After we prove some results to initial boundary value problems for no locally reacting boundaries. The results on the existence of solutions were obtained by using the constructive Faedo-Galerkin's method and compactness arguments. Results on the asymptotic behavior of solutions was given by Nakao's method. / Doutorado / Analise / Doutor em Matemática Aplicada