Peaked Waves

Gudrun talks to Anna Geyer. Anna is Assistant professer at TU
Delft in the Mathematical Physics group at the Delft Institute of
Applied Mathematics. She is interested in the behaviour of
solutions to equations which model shallow water waves.


The day before (04.07.2019) Anna gave a talk at the Kick-off
meeting for the second funding period of the CRC Wave phenomena
at the mathematics faculty in Karlsruhe, where she discussed
instability of peaked periodic waves. Therefore, Gudrun asks her
about the different models for waves, the meaning of stability
and instability, and the mathematical tools used in her field.


For shallow water flows the solitary waves are especially
fascinating and interesting. Traveling waves are solutions of the
form


representing waves of permanent shape f that propagate at
constant speed c. These waves are called solitary waves if they
are localized disturbances, that is, if the wave profile f decays
at infinity. If the solitary waves retain their shape and speed
after interacting with other waves of the same type, we say that
the solitary waves are solitons. One can ask the question if a
given model equation (sometimes depending on parameters in the
equation or the size of the initial conditions) allows for
solitary or periodic traveling waves, and secondly whether these
waves are stable or unstable. Peaked periodic waves are an
interesting phenomenon because at the wave crest (the peak) they
are not smooth, a situation which might lead to wave breaking.
For which equations are peaked waves solutions? And how stable
are they?


Anna answers these questions for the reduced Ostrovsky equation,
which serves as model for weakly nonlinear surface and internal
waves in a rotating ocean. The reduced Ostrovsky equation is a
modification of the Korteweg-de Vries equation, for which the
usual linear dispersive term with a third-order derivative is
replaced by a linear nonlocal integral term, representing the
effect of background rotation. Peaked periodic waves of this
equation are known to exist since the late 1970's. Anna presented
recent results in which she answers the long standing open
question whether these solutions are stable. In particular, she
proved linear instability of the peaked periodic waves using
semi-group theory and energy estimates. Moreover, she showed that
the peaked wave is unique and that the equation does not admit
Hölder continuous solutions, which implies that the reduced
Ostrovsky equation does not admit cusps. Finally, it turns out
that the peaked wave is also spectrally unstable. This is joint
work with Dmitry Pelinovsky.


For the stability analysis it is really delicate how to choose
the right spaces such that their norms measure the behaviour of
the solution. The Camassa-Holm equation allows for solutions with
peaks which are stable with respect to certain perturbations and
unstable with respect to others, and can model breaking waves.


Anna studied mathematics in Vienna. Adrian Constantin attracted
her to the topic of partial differential equations applied to
water waves. She worked with him during her PhD which she
finished in 2013. Then she worked as Postdoc at the Universitat
Autònoma de Barcelona and in Vienna before she accepted a tenure
track position in Delft in 2017.



References

A. Geyer, D.E. Pelinovsky: Spectral instability of the peaked
periodic wave in the reduced Ostrovsky equations, submitted
(arXiv)

A. Geyer, D.E. Pelinovsky: Linear instability and uniqueness
of the peaked periodic wave in the reduced Ostrovsky equation,
SIAM J. Math. Analysis, 51 (2019) 1188–1208

A. Geyer, D.E. Pelinovsky: Spectral stability of periodic
waves in the generalized reduced Ostrovsky equation, Lett. Math.
Phys. 107(7) (2017) 1293–1314

R. Grimshaw, L. Ostrovsky, V. Shrira, et al.: Long Nonlinear
Surface and Internal Gravity Waves in a Rotating Ocean, Surveys
in Geophysics (1998) 19: 289.

A. Constantin, W. Strauss: Stability of peakons, Commun. Pure
Appl. Math. 53 (2000) 603–610.

F. Natali, D.E. Pelinovsky: Instability of H1-stable peakons
in the Camassa-Holm equation, submitted (arXiv)




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