Qualitative analysis for the dispersion generalized Camassa-Holm equation

Official URL: https://risc01.sabanciuniv.edu/record=b3077562_(Table of contents)

In this thesis, we establish local well-posedness of the Cauchy problem for a dispersion generalized Camassa-Holm equation by using Kato’s semigroup approach for quasi-linear evolution equations. We show that for initial data in the Sobolev space Hs(R) with s > 7 2 +p, the Cauchy problem is locally well-posed, where p is a positive real number determined by the order of the differential operator L corresponding to the dispersive effect added to the Camassa-Holm equation. We first explain Kato’s semigroup approach on the Camassa-Holm equation and then give the proofs for the dispersion generalized Camassa-Holm equation. Finally, we compare the results of both equations and propose open problems related to the dispersion generalized Camassa-Holm equation.