The Well-posedness For A Class Of Shallow Water Equations

There are lots of nonlinear evolution equations depending on continuous time inmathematical physics equations which is the key to solving the equations of physics,fluid dynamics and other areas in many problems.In this paper, we mainly study some properties of solutions to the Cauchy problemof a class of shallow water wave equations arising from modern mechanics and physics.In Chapter1, we mainly introduced the background of shallow water waveequations, and some important results obtained by mathematicians in this field.In Chapter2, we deal with the generalized Camassa-Holm equation with bothquadratic and cubic nonlinearity. The local well-posedness of the solution is obtained bythe transport equations theory and the classical Friedrichs regularization method.In Chapter3, we consider a new two-component integrable system with cubicnonlinearity, and the local well-posedness of the solution is established.