Stability of smooth periodic travelling waves in the Camassa–Holm equation

We solve the open problem of spectral stability of smooth periodic waves in the Camassa–Holm equation. The key to obtaining this result is that the periodic waves of the Camassa–Holm equation can be characterized by an alternative Hamiltonian structure, different from the standard formulation common to the Korteweg-de Vries equation. The standard formulation has the disadvantage that the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We prove that the nonstandard formulation has the advantage that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region where the smooth periodic waves exist.