Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Aug. 14, 2008). PDF text: vi, 118 p. ; 460 K. UMI publication number: AAT 3297658. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaf 136). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--Oregon State University, 1991. / Typescript (photocopy). Includes bibliographical references (leaves 62-64). Also available on the World Wide Web.

In chapter 2 we use functional analytic methods and conservation laws to solve the initial-value problem for the Korteweg-de Vries equation, the Benjamin-Bona-Mahony equation, and the nonlinear Schrodinger equation for initial data that satisfy some suitable conditions. In chapter 3 we use the energy estimates to show that the strong convergence of the family of the solutions of the KdV equation obtained in chapter 2 in H³(R) as ε → 0; also, we show that the strong L²(R)-limit of the solutions of the BBM equation as ε → 0 before a critical time. In chapter 4 we use the Whitham modulation theory and averaging method to find the 2π-periodic solutions and the modulation equations of the KdV equation, the BBM equation, the Klein-Gordon equation, the NLS equation, the mKdV equation, and the P-system. We show that the modulation equations of the KdV equation, the K-G equation, the NLS equation, and the mKdV equation are hyperbolic but those of the BBM equation and the P-system are not hyperbolic. Also, we study the relations of the KdV equation and the mKdV equation. Finally, we study the complex mKdV equation to compare with the NLS equation, and then study the complex gKdV equation.

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.

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

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

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

In this study, we consider the generalized function solutions to nonlinear wave equation with distribution initial data. J. F. Colombeau shows that the initial value problem u_tt - Δu = F(u); m(x,0) = U_0; u_t (x,0) = i_1 where the initial data u_0 and u_1 are generalized functions, has a unique generalized function solution u. Here we take a specific F and specific distributions u_0, u_1 then inspect the generalized function representatives for the initial value problem solution to see if the generalized function solution is a distribution or is more singular. Using the numerical technics, we show for specific F and specific distribution initial data u_0, u_1, there is no distribution solution.