PDE Seminar (2015/10/5): A topological approach to an accumulation of eigenvalues of linearized operator along traveling waves

2015-10-5 16:30 - 2015-10-5 17:30
Faculty of Science Building #3, Room 309
Ayuki Sekisaka (Hokkaido University)
We consider the eigenvalue problem of traveling waves on bounded intervals. It is known that an accumulation of eigenvalues occurs when the size of bounded intervals tends to infinity. In particular, the set of accumulation points of eigenvalues characterizes the instability of waves on the large but bounded interval. The set of accumulation points is called the absolute spectrum because it characterizes the absolute instability of waves.

Our approach is based on the following topological theory. In the topological theory for a Strum-Liouville operator, the eigenvalue problem induces the Hamiltonian flow on the Lagrangian-Grassmannian, and each eigenfunction is characterized by a loop on this manifold. However, most linearized operators for traveling waves are non-self-adjoint operators. The obstacles were that the eigenvalues are not real and an induced flow has no longer Hamiltonian.

In this talk, we will show the asymptotic behavior of an induced flow on the Grassmannian, and give a topological proof of the accumulation of eigenvalues on the absolute spectrum using topological properties of the induced flow.