A. Geyer and D.E. Pelinovsky
Stability of smooth periodic traveling waves in the Degasperis-Procesi equation
Abstract:
We derive a precise energy stability criterion for smooth periodic waves in
the Degasperis–Procesi (DP) equation. Compared to the Camassa-Holm (CH) equation,
the number of negative eigenvalues of an associated Hessian operator changes in the
existence region of smooth perodic waves. We utilize properties of the period function
with respect to two parameters in order to obtain a smooth existence curve for the family
of smooth periodic waves of a fixed period. The energy stability condition is derived on
parts of this existence curve which correspond to either one or two negative eigenvalues of
the Hessian operator. We show numerically that the energy stability condition is satisfied
on either part of the curve and prove analytically that it holds in a neighborhood of the
boundary of the existence region of smooth periodic waves.
Keywords:
Camassa-Holm equation; Degasperis-Procesi equation; smooth periodic waves, existence and stability,
energy stability criterion.