Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 24.9.2015 at 14:00 Lecture room M-223

Olga Trichtchenko (University College London):

Comparison of stability of solutions to Hamiltonian water wave models

Abstract:

The goal of this work is to compare and contrast the stability results for solutions to different models for water waves.

It is known that high frequency instabilities exist for the nonlinear solutions to Euler's equations describing water waves [1, 2],

however not all models exhibit these instabilities. We will use a generalization of the theory used to predict existence of high

frequency instabilities in periodic Hamiltonian systems first proposed by MacKay [3] as well as Mackay and Saffman [4]

to see which water wave models meet the necessary conditions for these instabilities to arise. We will then examine

how these instabilities change if different conditions at the surface are included [5].

References

[1] B. Deconinck and K. Oliveras, The instability of periodic surface gravity waves. J. Fluid Mech., Vol 675 (2011), pp.141-167.

[2] B. Deconinck and O. Trichtchenko, Stability of periodic gravity waves in the presence of surface tension. European Journal of Mechanics

- B/Fluids, Vol 46 (2014), pp.97-108.

[3] R. S. MacKay, Stability of equilibria of Hamiltonian systems. Nonlinear Phenomena and Chaos, (1986) pp.254-270.

[4] R. S. MacKay and P. G. Saffman, Stability of water waves. Proc. Roy. Soc. London Ser. A, Vol 406 (1986), pp.115-125.

[5] P. A. Milewski, and Z. Wang Three dimensional flexural-gravity waves. Studies in Applied Mathematics, Vol 131 (2013), pp. 135148.