Browsing by Subject "Nonlinear waves"

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    Nonlinear mechanisms of wave propagation in periodic structures: harmonic generation, dispersion correction, and their interplay
    (2020-12) Jiao, Weijian
    Periodic structures have been extensively investigated due to their unique dynamical properties, which have enabled a broad range of practical applications, especially in the context of wave control. Considering nonlinearity in periodic structures not only leads to a more complete description, but also opens new doors for the design of functional and tunable metamaterials. In this thesis work, we are interested in the dynamical behavior of periodic structures in the weakly nonlinear regime. In the case of quadratic nonlinearity, a well-known effect on wave propagation is second harmonic generation (SHG), which gives rise to a secondary harmonic in addition to the fundamental harmonic that is nearly identical to the linear response. This effect provides an opportunity to nonlinearly activate a second harmonic that exhibits complementary modal characteristics to those of the fundamental harmonic, thereby enriching the modal characteristics involved in the total response. Then, this modal enrichment functionality is explored in 1D periodic structures featuring internal resonators via numerical and experimental analysis, in which we use SHG as a mechanism to achieve energy trapping and localization in the resonators. Moreover, we extend our focus to experimentally demonstrate all the key components induced by SHG in 2D lattices of repulsive magnets supported by pillars. As for cubic nonlinearity, the effect on wave propagation is an amplitude-dependent correction of the dispersion relation, which can manifest either as a frequency shift or as a wavenumber shift depending on how the excitation is prescribed. Compared to the vast study on frequency shift, the scenario of wavenumber shift has only been marginally explored. To fill this gap, we first present a multiple scales framework to analytically capture the wavenumber shift on the dispersion relation of monatomic chains, showing that wavenumber shift is associated with harmonic boundary excitation. We then extend the framework to periodic structures with internal resonators to achieve tunability of locally resonant bandgaps. Last, we investigate the effects of the interplay between quadratic and cubic nonlinearities in periodic waveguides. Through two conceptual applications, we demonstrate that these effects can be leveraged to unveil an array of wave control strategies for the design of tunable metamaterials with self-switching functionalities.
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    Universal dynamics of invasion fronts
    (2022-06) Avery, Montie
    This thesis focuses on understanding the spatiotemporal dynamics of instabilities in largephysical systems. The onset of instability, either through a change in system parameters or the introduction of an external agent, plays a central role in mediating state transitions and structure formation in many physical systems. Examples common to our every day experience include viral epidemics and invasive species in ecology. Dynamics in the wake of instability are often governed by an invasion process, in which localized perturbations to an unstable background state grow, spread, and select a new state in the wake of invasion. A fundamental question is to predict the speed of propagation and the selected state in the wake. The mathematical study of invasion processes began in the 1920s with the Fisher-KPP equation,a model for the spread of advantageous genes in biological populations. This started a long avenue of research into related equations, predicting propagation speeds by constructing appropriate super- and sub- solutions and controlling propagation using comparison principles. On the other hand, there is a substantial body of experimental and theoretical work in the physics literature, dating back to the plasma physics literature in the 1950s, providing universal predictions for invasion speeds based only on certain spectral stability properties. This universal guiding principle is known as the marginal stability conjecture. In this thesis, we give the first proof of the marginal stability conjecture in a model independentframework, in the case where the invasion process selects a spatially constant state in its wake. We expect the framework we develop to remain useful in predicting invasion speeds in pattern-forming systems. In the process of our proof, we develop new mathematical techniques for establishing stability estimates in the presence of essential spectrum, which we expect to have broader use in diffusive stability problems.