The Frobenius Integrable Decompositions Of The Nonlinear Evolution Equations And The Expanding Of The Integrable Systems

The major contents in this paper include:the Frobenius integrable decompositions of the nonlinear evolution equations and the expanding of the integrable systems. In the first chapter, the development of soliton theory and integrable system together with its research meaning are summarized. In the second chapter, firstly, the concept of the Frobenius integrable decompositions is introduced for partial differential equations. Secondly, a procedure is provided for determining a class of partial differential equations of polynomial type, which posses specified Frobenius integrable decompositons. Two concrete examples with logarithmic derivative Backlund transformations are given, and the presented partial differential equations are transformed into Frobenius integrable ordinary differential equations. Finally, the resulting solutions are illustrated to describe the solution phenomena. The third chapter can be divided into three parts. In the first part, the general theory and method of integrable system are introduced. In the second part, the super Lie algebra is constructed, by using of the designed isospectral problem and the supertrace identity, the super-Dirac soliton hierarchy as well as its super-bi-Hamiltonian structure are obtained. Finally, the integrable couplings of the Giachetti-Johnson(GJ) hierarchy are obtained by the perturbation approach and its Hamiltonian structure is given for the first time by component-trace identities.