Research On Exact Solutions Of(3+1)Dimensional Generalized Shallow Water Equation Based On Bilinear Method

The objective world essentially is complex and nonlinear.Nonlinear partial differential equations are widely used to explore and solve these complex nonlinear phenomena.Therefore,it is very important to find exact solutions of nonlinear equations for understanding the essential characteristics and motion rules of nonlinear systems.At present,methods for solving nonlinear equations have emerged in large numbers and have been rapidly developed,such as Darboux transfer-mation method,Backlund transformation method,Hirota bilinear method and so on.Based on Hirota bilinear method,this thesis deeply studies the intrinsic characteristics and various application techniques of bilinear structures.It constructs a class of Shallow Water wave(SWW)equations by linearization and combines symbolic computation to construct some exact solutions and the interaction solution.At the same time,their 3D images are drawn,which help to analyze the nature and shape of the solution.The thesis is arranged as follows:The first chapter briefly introduces the research background and current situation of soliton,some methods for solving nonlinear partial differential equations and the main work of this thesis.The second chapter briefly describes the definition,algorithm and basic properties of the Hirota bilinear method.The third chapter uses the Hirota bilinear method and the appropriate transformation formula to transform a class of SWW equations into a bilinear equation.With the idea ofquadratic function,the lump solution of the equation is obtained by symbolic calculation,and this lump is ensured by parameter constraints which is locally rational in any direction of each plane.Then the interaction solution of Lump-type soliton and single-striped soliton and the interaction solution of Lump-type soliton and kinked soliton are obtained respectively.By selecting appropriate parameters,the 3D images of the solutions are drawn to show their dynamic characteristics.Finally,using the double soliton solution and multi-soliton solution,the breather and several kinds of interaction solutions are obtained by the long-wave limit theory,and the mapping analysis are carried out.