The investigation of local weak solutions for a generalized Novikov equation

which has a matrix Lax pair [, ] and is shown to be related to a negative flow in the Sawada-Kotera hierarchy. Several conservation quantities and a bi-Hamiltonian structure were found in []. Himonas and Holliman [] applied the Galerkin-type approximation method to prove the well-posedness of strong solutions for Eq. () in the Sobolev space Hs(R) with s >  on both the line and the circle. Its Hölder continuity properties were studied in Himonas andHolmes []. The abstract Kato theoremwas employed in Ni and Zhou [] to show the existence and uniqueness of local strong solutions in the Sobolev space Hs(R) with s >  . The persistence properties of the strong solution were found. The local well-posedness for the periodic Cauchy problem of the Novikov equation in the Sobolev spaceHs(R) with s >   is done in Tiglay []. If the initial data are analytic, the existence and uniqueness of analytic solutions for Eq. () are obtained in []. It is worthy to mention that if the Sobolev index s≥  and sign conditions hold, the orbit invariants are applied to show the existence of periodic global strong solution. The scattering theory is used by Hone et al. [] to search for non-smooth explicit soliton solutions with multiple peaks for Eq. (). This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations (see [–]). In this work, we study the following generalized dissipative Novikov equation: