The Rational Solutions Of Higher Dimensional Shallow Water Equations Based On The Generalized Bilinear Operator

Soliton theory is an important part in nonlinear science. Its model,which shows the physical phenomenon, is widely used in different areas,such as physical oceanography, quantum mechanics, atmospheric science, and fluid mechanics.In particular, atmospheric blocking phenomenon (mKdV models), optical soliton in optical fiber communication (KdV model), the financial engineering of financial soliton (nonlinear Schrodi- nger equation) are modeled by soliton theory.Therefore, the study of integrability properties and the solutions for soliton equation can benefit a great number of areas. It is significant for both theoretical research and practical applications.This paper mainly discusses Hirota bilinear operator. We use Hirota bilinear method to get the rational solutions of the (2+1)-dimensional shallow water equation and the (2+1)-dimensional shallow water-like equation. Through the analysis of the rational solutions and images, we get its value in practical applications. The structure of this paper is shown below:The first chapter introduces the development of the soliton theory.Further, we discuss the common mathematical methods to solve the soliton equation. They are inverse scattering method, Backlund and Darboux transformation, and Hirota bilinear method.The second chapter consists of two parts. The first part mainly introduces the basic idea of Hirota bilinear method. Later, it discusses three transformation methods to convert the nonlinear partial differential equation into linear form, rational transformation, logarithmic transformation, and double logarithmic transformation. The second part discusses the Bell polynomial.The third chapter includes two parts. The first part introduces double linear operator. Through the variable transformation, we get the(2+1 )-dimensional bilinear form of shallow water equation. Then through the symbolic computation and transformation of coefficient, we get the rational solutions of (2+1 )-dimensional shallow water equation.The second part introduces the nature of the double linear. Then we compare the difference between generalized bilinear operator and classical bilinear operator. Through the symbolic computation, we get the generalized bilinear form of (2+1 )-dimensional shallow water-like equation. Through the Maple symbolic computation and the trans formation of coefficient, we get the rational solutions of the equation.