Stability of Periodic Waves in Nonlocal Dispersive Equations

Abstract

In this work consisting of joint projects with my advisor, Dr. Mathew Johnson, we study the existence and stability of periodic waves in equations that possess nonlocal dispersion, i.e. equations in which the dispersion relation between the temporal frequency, omega, and wavenumber, k, of a plane wave is not polynomial in ik. In models that involve only classical derivative operators (known as local equations), the behavior of the system at a point is influenced solely by the behavior in an arbitrarily small neighborhood. In contrast, equations involving nonlocal operators incorporate long-range interactions as well. Such operators appear in numerous applications, including water wave theory and mathematical biology. Specifically, we establish the existence and nonlinear stability of a special class of periodic bound state solutions of the Fractional Nonlinear Schrodinger Equation, where the nonlocality of the fractional Laplacian presents formidable analytical challenges and elicits the development of functional-analytic tools to complement the absence of more-understood techniques commonly used to analyze local equations. Further, we use numerical methods to survey the existence and spectral stability of small- and large-amplitude periodic wavetrains in Bidirectional Whitham water wave models, which implement the exact (nonlocal) dispersion relation of the incompressible Euler equations and are thus expected to better capture high-frequency phenomena than the unidirectional Whitham and Korteweg-de Vries (KdV) equations.