It is known that there are nonlinear wave equations with localized solitary wave solutions. Some of these solitary waves are stable (with respect to a small perturbation of initial data)and have nonzero spin (nonzero intrinsic angular momentum in the centre of momentum frame). In this paper we consider vector-valued solitary wave solutions to a nonlinear Klein-Gordon equation and investigate the behavior of these spinning solitary waves under the influence of an externally imposed uniform magnetic field. We find that the only stationary spinning solitary wave solutions have spin parallel or antiparallel to the magnetic field direction.

In this thesis, we prove the existence of large frequency periodic solutions for the nonlinear wave equations utt − uxx − v(x)u = u3 + [fnof]([Omega]t, x) (1) with Dirichlet boundary conditions. Here, [Omega] represents the frequency of the solution. The method we use to find the periodic solutions u([Omega]) for large [Omega] originates in the work of Craig and Wayne [10] where they constructed solutions for free vibrations, i.e., for [fnof] = 0. Here we construct smooth solutions for forced vibrations ([fnof] [not equal to] 0). Given an x-dependent analytic potential v(x) previous works on (1) either assume a smallness condition on [fnof] or yields a weak solution. The study of equations like (1) goes back at least to Rabinowitz in the sixties [25]. The main difficulty in finding periodic solutions of an equation like (1), is the appearance of small denominators in the linearized operator stemming from the left hand side. To overcome this difficulty, we used a Nash-Moser scheme introduced by Craig and Wayne in [10]. / text

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains ix, 134 p.; also includes graphics. Includes bibliographical references (p. 130-134). Available online via OhioLINK's ETD Center

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Thesis (Ph.D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains ix, 176 p.; also includes graphics. Includes bibliographical references (p. 172-176). Available online via OhioLINK's ETD Center

A periodic structure consists of spatially repeating unit cells. From man-made multi-span bridges to naturally occurring atomic lattices, periodic structures are ubiquitous. The periodicity can be exploited to generate frequency bands within which elastic wave propagation is impeded. A limitation to the linear periodic structure is that the filtering properties depend only on the structural design and periodicity which implies that the dispersion characteristics are fixed unless the overall structure or the periodicity is altered. The current research focuses on wave propagation in nonlinear periodic structures to explore tunability in filtering properties such as bandgaps, cut-off frequencies and response directionality. The first part of the research documents amplitude-dependent dispersion properties of weakly nonlinear periodic media through a general perturbation approach. The perturbation approach allows closed-form estimation of the effects of weak nonlinearities on wave propagation. Variation in bandstructure and bandgaps lead to tunable filtering and directional behavior. The latter is due to anisotropy in nonlinear interaction that generates low response regions, or "dead zones," within the structure.The general perturbation approach developed has also been applied to evaluate dispersion in a complex nonlinear periodic structure which is discretized using Finite Elements. The second part of the research focuses on wave dispersion in strongly nonlinear periodic structures which includes pre-compressed granular media as an example. Plane wave dispersion is studied through the harmonic balance method and it is shown that the cut-off frequencies and bandgaps vary significantly with wave amplitude. Acoustic wave beaming phenomenon is also observed in pre-compressed two-dimensional hexagonally packed granular media. Numerical simulations of wave propagation in finite lattices also demonstrated amplitude-dependent bandstructures and directional behavior so far observed.