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Modulational and Subharmonic Dynamics of Periodic Waves
Perkins, Wesley Robert
Perkins, Wesley Robert
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Abstract
The overarching theme of this dissertation is to investigate the stability and dynamics of spatially-periodic travelling and stationary wave solutions to partial differential equations (PDEs) arising from physical applications. First we study the modulational dynamics of interfacial waves rising buoyantly along a conduit of a viscous liquid. Formally the behavior of modulated periodic waves on large space and time scales may be described through the use of Whitham modulation theory. The application of Whitham theory, however, is based on formal asymptotic (WKB) methods, thus removing a layer of rigor that would otherwise support their predictions. In this first study, we aim at rigorously verifying the predictions of the Whitham theory, as it pertains to the modulational stability of periodic waves, in the context of the so-called conduit equation, a nonlinear dispersive PDE governing the evolution of the circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. Using rigorous spectral perturbation theory, we connect the predictions of Whitham theory to the rigorous spectral (in particular, modulational) stability of the underlying wave trains. We then study the linear dynamics of spectrally stable $T$-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\"{o}dinger equation with forcing that arises in nonlinear optics. It is known that such $T$-periodic solutions are nonlinearly stable to $NT$-periodic, i.e., subharmonic, perturbations for each $N\in\mathbb{N}$ with exponential decay rates of the form $e^{-\delta_N t}$. However, both the exponential rates of decay $\delta_N$ and the allowable size of initial perturbations tend to 0 as $N\to\infty$ so that this result is non-uniform in $N$ and is, in fact, empty in the limit $N=\infty$. We introduce a methodology in the context of the LLE by which a uniform stability result for subharmonic perturbations may be achieved at the linear level. The obtained uniform decay rates are shown to agree precisely with the polynomial decay rates of localized, i.e., integrable on the real line, perturbations of such spectrally stable periodic solutions of the LLE. A key component of the proofs in this study is the introduction of space-time dependent modulations, which, as we show, allows us to justify Whitham theory at the level of linear dynamics for the LLE. This work both unifies and expands on several existing works in the literature concerning the stability and dynamics of such waves, and sets forth a general methodology for studying such problems in other contexts. Interestingly, we were unable to push the above analysis to the nonlinear level due to an unavoidable loss of derivatives that occurs in our iteration scheme. If the PDE has dissipation in the highest-order term, one can regain these lost derivatives through a technique known as ``nonlinear damping.'' Unfortunately, the dissipation in the LLE occurs in the lowest-order term, leaving no hope of regaining derivatives through a nonlinear damping estimate. As a proof of concept, we explore if, in the presence of a nonlinear damping estimate, the methodology we developed for the linear analysis could be used to develop an analogous nonlinear subharmonic stability result which is uniform in $N$. To that end, we investigated the stability and nonlinear dynamics of spectrally-stable wave trains in reaction-diffusion systems. Using the nonlinear damping estimate present in reaction-diffusion systems, we were able to successfully introduce a methodology by which a stability result for subharmonic perturbations which is uniform in $N$ may be achieved at the {\it nonlinear} level. This proof of concept therefore motivates the idea that methodologies for overcoming the loss of regularity that occurs in {\it localized} nonlinear iteration schemes can be modified to similarly overcome the loss of regularity that occurs when establishing nonlinear subharmonic stability results that are uniform in the period of perturbation. Consequently, in the final chapter of the dissertation we develop a new methodology that allows us to circumvent the loss of regularity in the case of localized perturbations and in the presence of weak damping, i.e., in the absence of a nonlinear damping estimate. We return to our study of the Lugiato-Lefever equation and consider the nonlinear stability of spectrally stable periodic stationary solutions of the LLE. In our first study of the LLE, we used a delicate decomposition of the associated linearized solution operator to obtain linear stability results to localized perturbations with polynomial rates of decay to a spatio-temporal phase modulation of the underlying wave. In this study, we present a new nonlinear iteration scheme in which the aforementioned loss of derivatives is compensated through a coupling to a separate ``unmodulated'' iteration scheme in which derivatives are not lost, yet where perturbations decay too slow to close an independent iteration scheme. Our work establishes the nonlinear stability of spectrally stable periodic stationary solutions of the LLE to localized perturbations with precisely the same polynomial decay rates predicted from the linear theory. Moreover, as our study of reaction-diffusion systems showed, it motivates the methodology by which we may obtain a nonlinear subharmonic stability result for the LLE that is uniform in the period of perturbation, a result that is currently under investigation by the author et al.
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Date
2021-05-31
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University of Kansas
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Keywords
Mathematics, Applied mathematics, Dynamics, Nonlinear Waves, Periodic Waves, Stability