A note on the analytic solutions of the Camassa-Holm equation

In this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces. Using theAbstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a unique analyticsolution. Moreover, if the initial data is real analytic, belongs to H s (R) with s > 3/2, u0 L1 < ∞ and u0 − u0xx does not changesign, we prove that the solution stays analytic globally in time.