Stability of smooth periodic traveling waves in the Degasperis–Procesi equation

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 periodic 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 with 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.