The Research On Integrability Of Soliton Equations Based On The Bell Polynomial Method

This dissertation mainly consists of two parts. The first one is to study the exact solutions, integrability properties and integrable constraint con-ditions of some soliton equations by using the Bell polynomial method and the Hirota bilinear method. The second one is to investigate the in-tegrability properties of some supersymmetry equations by the super Bell polynomial approach.Concretely, the third chapter mainly apply Bell polynomial approach to study integrability problem of three soliton equations. First of all, us-ing the Bell polynomial approach to obtain the bilinear form with an ex-tra auxiliary variable, bilinear Backlund transformation, Lax pair and in-finite conservation laws for the Lax fifth-order KdV (Lax fKdV) equa-tion, then it can be said that this equation is completely integrable. Sec-ondly, for the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov (ANNV) system, firstly the bilinear form with an arbitrary function φ(y) is obtained by the Bell polynomial approach. Furthermore, the new ex-act solutions with an arbitrary function in y are presented based on the Hirota bilinear method, and soliton interaction properties are discussed by the graphical analysis. Besides, the bilinear Backlund transformation and the corresponding Lax pair are also derived. Finally, for the gener-alized (2+1)-dimensional variable coefficients KdV equation, the bilinear Backlund transformation. Lax pair and infinite conservation laws are ob-tained under certain constraints. Therefore this equation turns out to be completely integrable under certain constraints. In the fourth chapter, through the relation between the super Bell poly-nomial and the super Hirota bilinear operator, the bilinear form, bilinear Backlund transformation, Lax pair and infinite conservation laws are de-rived for the supersymmetric Ito equation. Further extends the application of the Bell polynomial.