### The parabolic resonance instability

Vered Rom-Kedar

The parabolic resonance instability was first identified a decade ago
when we studied near-integrable two degrees of freedom Hamiltonian
systems in which the angular momentum is nearly preserved [1,2]. We
then showed that it appears persistently in near integrable n d.o.f.
Hamiltonian families depending on p parameters provided n+p???3 ,
namely, that it is a ubiquitous instability [3]. Since then, we
observed parabolic resonances in various applications, such as the
forced periodic 1D NLS equation and the driven surface waves. For the
forced 1D NLS we have recently shown that this instability can lead to
spatial decoherence of small amplitude nearly flat solutions [4,5].
The analysis of the parabolic resonance instability turns out to be
elegant and revealing: we propose that it is governed by adiabatic
chaos and that the phase-space volume of its chaotic zone scales
differently from both the elliptic and the hyperbolic cases [6].
[1] V. Rom-Kedar; Parabolic resonances and instabilities, Chaos,
7(1):148-158, 1997.
[2] V. Rom-Kedar and N. Paldor; From the tropic to the poles in
forty days. Bull. Amer. Mete. Soc., 78(12):2779-2784, 1997.
[3] A. Litvak-Hinenzon and V. Rom-Kedar; On energy surfaces and
the resonance web; SIAM J. Appl. Dyn. Syst. 3(4), 525---573, 2004.
[4] E. Shlizermann and V. Rom-Kedar; Parabolic Resonance: A
Route to Hamiltonian Spatio-Temporal Chaos, Phys Rev Letters, 102,
033901, 2009.
[5] E. Shlizermann and V Rom-Kedar; Three types of chaos in the
forced nonlinear Schrodinger equation. Physical Review Letters, 96,
024104, 2006.
[6] V. Rom-Kedar and D. Turaev; The Parabolic resonance
instability, Nonlinearity, to appear, 2010.