The Study Of Several Soliton And Supersymmetric Equations

The discussion of integrability of soliton equation is an important sub-ject in soliton theory. The Painleve analysis has shown its usefullness in studying of nonlinear evolution equations (NLEEs), especially in the sense of integrability. There are some close connections between integrable mod-els and Painleve property. To find exact solutions of nonlinear soliton equa-tions is another essential aspect in the field of soliton. Thus, a variety of distinct methods have been developed. Among them, the Hirota bilinear method and the Backlund transformation are proved to be the effective ap-proaches in search of exact solutions for NLEEs. In this dissertation, we first present the explicit Painleve test for a new (2+1)-dimensional general-ized KdV equation. Then some exact solutions of the equation are obtained based on the Hirota bilinear method. In addition, the Backlund transfor-mations for two (2+1)-dimensional breaking soliton equations have been generated in bilinear form.As we know that the system of integrable models has the wide range of applications. And accordingly, it is worthwhile searching their supersym-metric equations. A number of well known integrable equations have been generalized into the supersymmetric context and many of the tools used in standard theory have been extended to this framework. In the last part of this thesis, on the basis of reference to relevant articles, we outline the applications of the Hirota bilinear method in supersymmetric equations.In details,In chapter2, the new (2+1)-dimensional generalized KdV equation which exists the bilinear form is mainly discussed. We prove that the equation does not admit the Painleve property even by taking the arbitrary constant a=0. In addition, based on Hirota bilinear method, periodic wave solutions in terms of Riemann theta function and rational solutions are derived respectively. The asymptotic properties of the periodic wave solutions are analyzed in detail.In chapter3, by virtue of some bilinear identities, the bilinear Backlund transformations for the Calogero equation and Bogoyavlenskii breaking soliton equation are obtained. It is a bilinear Backlund transformation for Bogoyavlenskii breaking soliton equation that has two-parameters. By virtue of the second bilinear Backlund of the Calogero equation derived by its Lax pair, the N-soliton solutions are constructed with a purely algebraic procedure.In chapter4, taking the KdV equation for example, we discuss how to seek supersymmetric KdV equation. And then, we study how to derive the bilinear form and the soliton solutions for supersymmetric KdV equation.