Some Study Of Integrable Wave Equations

In this thesis, we investigate to three kinds of integrable wave equations.First, we consider the well-known Camassa-Holm equation on the real line. We give a new and direct proof for McKean's theorem on wave breaking of this equation (how to blow up). The blow-up profile is also analyzed. Meanwhile, an algebraic decay rate of the strong solution to this equation in L∞-space is also proved.Then, we intend to study a class of nonlocal dispersive models-theθequations which arises in shallow water theory. We improve previous results and get some new criteria on blow-up, then discuss the global existence of the solution and establish sufficient conditions on the propagation speed for them.Finally, for a new integrable equation (we call it the Novikov equation in this thesis), we prove that the Cauchy problem for this equation is locally well-posed in the Besov spaces B2,τs with the critical index s=3/2. Then, we establish sufficient conditions on the initial data to guarantee the formulation of singularities in finite time. A global existence result is also found.