The local well-posedness for the dispersion generalized Camassa-Holm equation

In this paper, we establish the local well-posedness of the Cauchy problem for a dispersion generalized Camassa-Holm equation. The present generalization is obtained by replacing the operator $ (1-\partial _{x}<^>2) $ (1-partial derivative x2) of the Camassa-Holm equation with $ (1-L\partial _{x}<^>2) $ (1-L partial derivative x2) where L is a positive differential operator with order p, an even positive integer. We follow Kato's semigroup approach for quasi-linear evolution equations but use L-dependent operators and norms. We show that the Cauchy problem is locally well-posed in a Banach space for which the norms are equivalent to Sobolev space norms and the regularity index has a threshold depending on p. Considering the special cases of the operator L, we verify that our results are consistent with those presented in the literature.