The stability of nonlinear waves and the Maslov index

A critical concern for distinguished solutions in evolutionary PDEs is their stability, that is, their robustness to perturbations in initial conditions. Indeed, this determines (at least in theory) whether the solution – which represents a state of the physical system that the PDE models – is physically realisable or not. In this talk, I’ll discuss how the Maslov index – an intersection index from symplectic geometry – can be used to understand the weakest notion of stability, known as spectral stability, for nonlinear waves in 1+1 PDE’s. In particular, the Maslov index is used to detect the unstable eigenvalues of the temporal spectral problem associated with the linearisation of the system about the solution of interest. Arguably, the simplest application of the Maslov index lies in the proof of the Sturm oscillation theorem, which states that the eigenfunction for the nth eigenvalue of a second-order selfadjoint operator has n-1 zeros. To set the stage, I will discuss a proof of Sturm’s theorem, the main idea being that one can glean spectral information from the geometric structure of an eigenfunction. Next, I’ll show how these ideas can be generalised to study the spectral stability of standing wave solutions to a fourth order nonlinear Schrödinger equation with cubic nonlinearity. The main result is a lower bound for the number of real unstable eigenvalues of the linearised operator.

This is based on recent work with Robby Marangell.