Integrable Systems In The Sense Of Lax Pair And Painlevé Property And Exact Solutions

Soliton theory is one of the important branches of nonlinear science. It has crescent application in many fields. In the field of soliton theory, integrability of the nonlinear equations is of vital important research problem all the time. The more integrable equations not only facilitate to solve the problem of identification integrability in the theory, but also provide lots of valuable nonlinear evolution equations.In this paper, we use the asymptotically Fourier reduction method and obtain a new integrable equation in the sense of Lax pair from the mKP equation. In Ref[68], Lou has pointed out that starting from a conformal invariant form to find a high dimensional integrable model is a convenient approach. We obtain the new high dimensional integrable systems from Schwartz form of the Boussinesq equation with this idea. The single soliton solution and the travelling waves solutions for arbitrary dimensional integrable systems are obtained by the Painlevé-B(a|¨)cklund transformation.In addition, to find the exact solutions of nonlinear evolution equations is valuable and significant in fact. A large number of useful methods have been proposed to construct soliton solutions. Among these methods, the Darboux transformation is the powerful method to construct solutions. In this thesis, We try to find the Darboux transformation for the Wu-Zhang(WZ) equation and two-component Camassa-Holm(2-CH) equation. We investigate muti-soliton solutions and soliton like solutions of the WZ equation by the Darboux transformation and two-soliton head on and overtaking collisions. These collisions are not elastic because the shape, phase and velocity of the soliton (or kink) are changed. We find a single loop solution and two loop solution, muti-soliton (muti-soliton like) solutions of the 2-CH by applying for the Darboux transformation and investigate property of muti-soliton propagation. The interaction of two solutions is inelastic and elastic with electing different seed solutions.