Analytic Study Of Some Nonlinear Evolution Equations With Solitons And Integrable Property

Nonlinear evolution equations (NLEEs) can be used to simu-late many complicated phenomena in such fields as biology, plasma physics and fluid mechanism. Especially, as one integrable property of the NLEE, solitons attract the researchers’attention for their roles in explaining those complex phenomena. With the development of symbolic computation, some analytic methods are presented for solv-ing the NLEE, including Hirota bilinear method, Bell-polynomial ap-proach, Wronskian determinant and Backlund transformation. Based on symbolic computation, this dissertation analytically investigates some NLEEs. The work of this dissertation includes the following three aspects:(1)Under investigation in this paper is a Kadomtsov-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation, which models the de-formation of a compressible hyperelastic plate to a uniformly pre-stressed state. With the aid of the Hirota method and symbolic computation, the bilinear form and N-soliton solutions for the KP-BBM equation are derived. Via the Bell-polynomial approach, the Backlund transformation is constructed. Based on the bilinear form and Backlund transformation, we obtain the Wronskian solutions for the KP-BBM equation. Meanwhile, propagation characteristics and interaction behaviors of the solitons are discussed through the graph-ical analysis. Bell-shape solitons are obtained and the interactions are proved to be elastic through the asymptotic analysis.(2) Recently, Bell-polynomial approach is employed to obtain Back lund transformations and Lax pairs for nonlinear evolution equations in a direct way. By taking variable-coefficient Boussinesq and mKdV equations as examples, we obtain their Bell-polynomial expressions and Bell-polynomial-typed Backlund transformations, which can cast, respectively, into the bilinear forms and bilinear Backlund transfor-mations. In addition, the corresponding inverse scattering equations or Lax pairs are derived by linearizing their Bell-polynomial-typed Backlund transformations.(3)In this part, the Sawada-Kotera equation with a nonvanish-ing boundary condition is studied, which describes the evolution of steeper waves of shorter wavelength than those described by the Korteweg-de Vries equation does. With the binary-Bell-polynomial, Hirota method and symbolic computation, the bilinear form and N-soliton solutions for this model are derived. Meanwhile, propaga-tion characteristics and interaction behaviors of the solitons are dis-cussed through the graphical analysis. Via Bell-polynomial approach, the Backlund transformation is constructed in both the binary-Bell-polynomial and bilinear forms. Based on the binary-Bell-polynomial-type Backlund transformation, we obtain the Lax pair and conserva-tion laws associated.