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.

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