Stability of stationary periodic solutions of nonlinear wave equations

The stability of solitary wave solutions of nonlinear wave equations has been extensively studied over the last few decades. Recently, there has been increasing interest in studying the stability of the periodic counterparts of these solitary wave solutions. This has been facilitated by the development of new numerical techniques, as well as an improved understanding of the analysis required to approach these questions. Some of each will be discussed in this talk.

I will discuss Hill's method, which is a numerical method to compute spectra of linear operators, and convergence results we have been able to prove for it. Then I will discuss the stability of periodic solutions of integrable equations such as the Korteweg-de Vries and nonlinear Schrödinger equations, emphasizing the complete results that may be obtained.