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.
