Fall 2024 Schedule
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Stability of Nonlinear Waves in Hamiltonian PDE
Nonlinear dispersive wave equations arise as reduced mathematical models from governing equations of mathematical physics, such as the Navier-Stokes and Maxwell equations. These reduced models combine the leading-order balance between nonlinear and dispersive effects present in wave propagation. The existence and stability of coherent structures such as traveling, standing, or periodic wave solutions, and the long-time dynamics near these coherent structures are of great importance. In this talk, I will present some of the ideas (ODE and PDE theory) that one typically employs to define and study the stability of such structures. I will also discuss the models I have investigated the existence and dynamics of nonlinear waves.
3:00 pm, On Zoom
$L^2$ theory for compressible fluid
We will introduce the general $L^2$ theory for compressible fluid. There are two settings for this method: the inviscid one and the viscous one. Today we will give details on the viscous setting for the $L^2$ theory. We will also briefly discuss the ongoing project with Yannan Shen on the stability of dispersive shock.
3:00 pm, 256 Snow Hall
Modulational Instability of Small Amplitude Periodic Traveling Waves in the Novikov Equation
We study the spectral stability of smooth, small-amplitude periodic traveling wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. Specifically, we investigate the L2(ℝ)-spectrum of the associated linearized operator, which in this case is an integro-differential operator with periodic coefficients, in a neighborhood of the origin in the spectral plane. Our analysis shows that such small-amplitude periodic solutions are spectrally unstable to long-wavelength perturbations if the wave number if greater than a critical value, bearing out the famous Benmajin-Feir instability for the Novikov equation. On the other hand, such waves with wave number less than the critical value are shown to be spectrally stable. Our methods are based on applying spectral perturbation theory to the associated linearization.
3:00 pm, 306 Snow Hall
Steady States and Dynamics for Nematic Liquid Crystals
We begin by considering a nematic liquid crystal placed between two parallel plates. Within this set-up we use the Ericksen-Leslie model with certain boundary conditions to describe this physical system. From this model, we will first look into steady states in the system which already provides us with a variety of cases to investigate. We then study the stability of these steady states using various methods. These steady states fall into a variety of cases which the study of shows a variety of dynamics including the presence of a saddle node bifurcation and further dynamics of the unstable states shows the evidence of the existence of heteroclinic orbit.
3:00 pm, 306 Snow Hall
Initial-boundary value problems for nonlinear Schrödinger equations
We will discuss the Hadamard well-posedness (existence and uniqueness of solution; continuity of the data-to-solution map) of certain nonlinear dispersive partial differential equations of Schrödinger type in the context of initial-boundary value problems. Such problems arise naturally in applications where the spatial domain involves a boundary. In these cases, apart from the usual initial conditions that are present in initial value (Cauchy) problems, it is necessary to also prescribe appropriate (nonzero) boundary conditions. We will focus on how these boundary conditions affect the well-posedness theory, importantly in regard to the method and techniques that must be specifically developed for establishing such a theory in the initial-boundary value problem setting.
3:00 pm, 306 Snow Hall