Radius of analyticity for the Camassa-Holm equation on the line

Using estimates in Sobolev spaces we prove that the solution to the Cauchy problem for the Camassa-Holm equation on the line with analytic initial data u0(x) and satisfying the McKean condition, that is the quantity m0(x)=(1−∂x2)u0(x) does not change sign, is analytic in the spatial variable for all time. Furthermore, we obtain explicit lower bounds for the radius of spatial analyticity r(t) given by r(t)≥A−1(1+C1Bt)−1exp{−C0‖u0‖H1t}, where A,B,C1 andC0 are suitable positive constants.