| In Chapter I, we will introduce some background of the shallow water waves theory, and derive the Bona-Chen-Lannes system from the Euler equations by the asymptotic expansion. It is easy to see that in one dimension the above system is just the same as the Bona-Chen-Saut system. At the end of this chapter, some recent studies on these systems are given in brief.Solitary wave solutions of the BCL system are given in Chapter II. Firstly the BCL system is reduced to a nonlinear nonhomogeneous ordinary differential equations by an ansatz on the solution to the BCL system. Under the assumption that the perpendicular component of solution to propagation of wave is stationary, Some solitary wave solutions are given by M.Chen's Method. Without the above assumption, the reduced system is so complicated that more studies should be needed.In Chapter III, under the condition that guarantees the BCL system have a conservation law, the global well-posedness of the BCL system are obtained by extension of locally solution. However, a restriction of initial values that must be suitable small in the Sobolev norms, is imposed because of the non-positive-definite of the conservation law.In Chapter IV, well-posedness of the BCL system is studied without conservation law. However, the BCL system with viscous term is considered since the decay of the linearized BCL system is not enough. The system is diagnosed and an equivalent system is given by integral operators. We will use the techniques such as harmonic analysis, pseudo-differential operator etc., to obtain the necessary estimates. At last, by construction of functions spaces, the global well-posedness is obtained under the assumption of small initial values.In Chapter V, the global well-posedness of the BCL system with visous term is obtained in the degenerate case. This is an extension of Chapter IV.In Chapter VI, the regularities of the solutions to the above systems studied in the former chapters are given by the bootstrap method.
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