Painleve analysis and exact solutions of the Boussinesq-Burgers equation
|Keywords||Boussinesq-Burgers equation Painleve test resonances Darboux -Backlund transformation Schwarzian derivative equation|
Painleve analysis is a useful method in soliton theory. It can be used to prove integrability and find solutions of nonlinear partial differential equations.In this thesis, the (1+1) and (2+1) dimensional Boussinesq-Burgers equations(B-B equation) are studied by means of the Painleve analysis. Some Darboux-Backlund transformations are obtained. The main results are arranged as following.(1). Painleve test is used to study the (1+1) dimensional B-B equationFour branches are obtained from the Painleve analylsis. The branches are studied one by one. Their Painleve propertys are proved and Darboux-Backlund transformations are obtained. Through Darboux-Backlund transformations, some properties are studied such as Schwarzan derivative equation, soliton solutions and so on.(2). The Painleve property of (2+1) dimensional B-B equation is studied.Its Painleve integrability is proved and its soliton solutions are obtained. The Darboux -Backlund transformation is found, and bilinear transformation is obtained by means of the Painleve analysis. The solitons of B-B equation are obtained by the bilinear transformation. Interestingly, we find that the solutions obtained by the both methods are the same to a certian extent.(3). Soliton fusion or fission phenomenon of 2-soliton solutions are found by graphics analysis. The emergence of fusion phenomenon is revealed by the asymptotic analysis.