On the Cauchy Problem for the Fractional Camassa-Holm Equation

In this paper, we consider the Cauchy problem for the fractional Camassa-Holm equation which models the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. Using Kato’s semigroup approach for quasilinear evolution equations, we prove that the Cauchy problem is locally well-posed for data in $H^s(\mathbb{R})$, $s>\frac{5}{2}$.