Hirota Direct Method And Bell Polynomials Approach On The Soliton Equations

This dissertation mainly consists of two parts. The first part is to present some exact solutions to the (2+1) dimensional soliton equation. In the second part, the Bell polynomials are applied to solve the problem of integrability of the BLMP equation and non-isospectral KP equation.The third chapter first derives the bilinear form of the (2+1)-dimensional soliton equation and obtains its N-soliton solution in Wronski form. Then it presents the Grammian solution. At the same time, by using the variable separation method, some other kinds of exact solutions are generated. Then periodic wave solutions and dromion solutions are presented by choosing some special functions.The fourth chapter mainly concerns the application of the Bell poly-nomials. First the transformation is established between the Bell polyno-mials and the bilinear Backlund operator. Then according to the meaning of Bell polynomials, the constraint condition is established, so that the two solutions to the equation are obtained. What's more, by transformation, the BLMP equation and non-isospectral KP equation are also obtained. Based on the above work, Lax-pair and the infinite conservation laws to the BLMP equation and non-isospectral KP equation are also obtained with the binary Bell polynomials approach, from which it is concluded that these equations are completely integrable.