Seminar on Qualitative Theory of Differential Equations - Richard Kollár (22.11.2018)
Thursday 22.11.2018 at 14:00, Lecture room M/223
Richard Kollár:
Spectral Stability of Small Amplitude Periodic Waves for High-Order KdV Equations
Abstract:
The holy grail of the stability theory of nonlinear waves would be an explicit formula that would allow to determine the stability or instability of a wave. We present results that form a step in this direction. The spectral stability of small-amplitude periodic waves in scalar Hamiltonian problems can be studied as a perturbation of the zero-amplitude case. A necessary condition for stability of the wave is that the unperturbed spectrum is restricted to the imaginary axis. Instability can come about through a Hamiltonian-Hopf bifurcation, i.e., of a collision of purely imaginary eigenvalues of the Floquet spectrum of opposite Krein signature. In recent work on the stability of small-amplitude waves the dispersion relation of the unperturbed problem was shown to play a central role. We demonstrate that the dispersion relation provides even more explicit information about wave stability: the sign of the product of the Krein signatures of two colliding eigenvalues to detect possible instabilities of non-zero amplitude waves turns is the root of an explicitly constructed polynomial of half the degree of the dispersion relation that can be studied using approaches from number theory, revealing a quite unexpected connection. Throughout we use balanced higher-order KdV equations as an example.