The Studies Relative To Some Types Of Water-wave Equations

In this paper, we study some problems relative to some types of water-wave models, which are variable depth shallow water equation, rotational-two-component camassa-holm system, instability of equatorial edge waves in stratified flow and steady periodic rotational gravity waves with negative surface tension. This thesis, construct-ed by five parts, is organized as follows:In chapter 1, the backgrounds of our investigations are introduced and some pre-liminary lemmas, which are useful throughout this thesis, are provided.In chapter 2, we consider a nonlinear evolution equation for surface waves in shallow water over uneven bottom. The local well-posedness in Sobolev space Hs(S) with s>3/2 is established by applying Kato’s theory. Besides, the local well-posedness in Besov space B2,13/2 is also obtained. Then the blow up criterion is de-termined in Hs(S), s> 3/2 and some blow-up results are given for a simplified model. Finally, persistence properties on strong solutions are investigated.In chapter 3, a modified two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid is derived, which is a model in the equato-rial water waves. The effects of the Coriolis force caused by the Earth’s rotation and nonlocal nonlinearities on blow-up criteria and wave-breaking phenomena are then investigated. Our refined analysis relies on the method of characteristics, conserved quantities and the local structure of the dynamics and is proceeded with the Riccati-type differential inequality. Besides, conditions which guarantee the permanent waves are obtained by using a method of the Lyapunov function. Next, we classify various localized solitary-wave solutions for the rotation-two-component Camassa-Holm sys-tem. In addition to those smooth solitary-wave solutions, we demonstrate that there are solitary waves with singularities, like peakons and cuspons, according to the rotat-ing parameter and the balance index in fluid convection between nonlinear steepening and amplification due to stretching. Finally, we show that horizontally symmetric weak solutions of this model must be travelling waves.In chapter 4, we first present an explicit exact solution to the edge wave problem in stratified geophysical flows with an underlying longshore current. Then we ana-lyze the short-wavelength perturbation approach for barotropic incompressible fluids. Finally, we prove, by applying this method to geophysical equatorial edge waves in stratified water, that these waves are unstable when their steepness exceeds a specific threshold.In chapter 5, we consider the two-dimensional steady periodic rotational gravity waves with negative surface tension. Local curves of small amplitude solutions of the resulting problem are obtained by using the Crandall-Rabinowitz local bifurcation theory. By means of the global bifurcation theory combined with the Schauder theory of elliptic equations with the Venttsel boundary conditions, the curves of small am-plitude solutions is extended to the global continuum of solutions. Furthermore, it is shown that those waves are necessarily symmetric about the crest under the assump-tion that their surface profiles are monotonic between troughs and crests and locally strictly monotonic near the troughs.