Seminar 22.11.2018: Richard Kollar

Seminar on Qualitative Theory of Differential Equations
organized by P.Quittner, M.Fila and R.Kollar

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Seminar 22.11.2018: Richard Kollar

Postby quittner » Fri Nov 16, 2018 2:17 pm

Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations

Thursday 22.11.2018 at 14:00 Lecture room M-223

Richard Kollár (KAMŠ FMFI UK):
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.
quittner
 
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