Research On The Exact Solutions And Integrability Of Three Soliton Equations

The dissertation focuses on the exact solutions and the integrability properties of three soliton equations. Firstly, based on the connection between the Hirota D-operators and the binary Bell polynomials, we discuss the integra-bility properties of the generalized Caudrey-Dodd-Gibbon-Kaeada equation and the generalized2+1-dimensional Calogero-Bogoyavlenskii-Schiff equation by the bilinear bell polynomial approach. We derive the corresponding bilinear repre-sentations and the Backlund transformations in bilinear forms for these two gen-eralized equations. Then by introducing the Hopf-Cole transformation, we get their Lax pairs respectively, from which we conclude both of them are integrable under certain constraints. The second part studies the generalized Zakharov e-quation by means of the first integral method. Lots of new exact solutions that never presented previously are obtained with the help of Mathematica.