This talk will focus on spectral stability of small-amplitude, periodic solutions to Hamiltonian, dispersive partial differential equations. In particular, it has been shown in the past that periodic travelling wave solutions to the full Euler equations describing inviscid, incompressible fluid flow, exhibit high frequency instabilities. However, some simpler model equations frequently used, do not. We will examine the nature of these instabilities, how they arise, and present a general condition for instability. In special cases, this condition reduces to considering the interval in which there are roots of a polynomial half the degree of the polynomial describing the dispersion relation. We will illustrate the method for computing spectral stability by considering solutions to the Korteweg-de Vries, Kawahara, Whitham and Boussinesq-Whitham equations.
Please access the talk through the Research Seminars page ( https://researchseminars.org/seminar/MO_Analysis ); you will need to register for a free account. If you have any difficulty, please contact Samuel Walsh (email@example.com).