Several Mathematic Research Methods Of The Nonlinear System

The soliton theory is an frontier disciplines,involving multiple disciplines and fields.Its research means and methods constitute differential equation and power systems,the classic analysis and functional analysis,Lie group and Lie algebra,complex analysis, topology, elliptic function etc discipline theory.At present,Solving nonlinear differential equation has the following methods in the soliton theory:anti-scattering method,The homogeneous balance method, Jacobi elliptic function method, bilinear method, Darboux transformation Backlund transform method etc.The contents of this paper mainly include using Jacobi elliptic function method solving the Jaulent - Miodek soliton equation,Using Darboux transformation method solving the generalized Mkdv equations,Using bilinear method respectively solving the soliton Caudrey - Dodd - CDGK Gibbon - Kaeada equation and the (2 + 1) -dimensional Boussinesq-Burgers equation.Chapter 1 mainly introduces the involved development history, the recent theoretical research characteristics, three kinds of mathematical calculation method and this paper topics and main work.Chapter 2 puts forward generalized Jacobi elliptic function expansion method.It has the wide range of application, not only can be used to structe nonlinear evolution equations or coupling nonlinear evolution equations of periodic wave solutions, but also in some cases can give corresponding solitary wave solution and the periodic solution.To solve the soliton equation Miodek equations Jaulent - for example to detailed Jacobi elliptic functions extended demonstration expansion method. State detailly generalized Jacobi elliptic functions expansion method taking the solves of the soliton equation Jaulent - Miodek equationsas an example .Chapter 3 introduces N times Darboux transformation method structing the exact solution of the equation.This paper constructs Jaulent - Miodek equations N times of cloth transform, and gains the accurate soliton solution.Chapter 4 researchs Hirota bilinear method, and probes into the commonly used transformation method of the nonlinear partial differential equation from linear equations into bilinear equations : rational transformation, bi-logarithm transformation.Take respectively solving the soliton Caudrey - Dodd - CDGK Gibbon - Kaeada equation and the (2 + 1) -dimensional Boussinesq-Burgers equation as example.