The Study Of Nonlinear Ordinary Differential Equations And Solitons

With the development of the Nonlinear Science, lots of Nonlinear Evolution Equations (NLEEs) are found, and they are playing important roles in many different physics fields. Seeking the solutions of NLEEs is now a very important subject in Nonlinear Science. There is not a systemic and uniform method for all NLEEs, though some exact solutions have been found, which just for a few kinds of equations. In this case, it becames importandly to study the NLEEs. The soliton solutions are one kind of significative solutions because of its important applications. In this paper, several kinds of significant methods are studied to solve the NLEEs, and at the same time, some new one-soliton solutions and multi-soliton solutions of NLEEs are given.The basic content of this paper are given as following: Chapter one is devoted to reviewing the history and development of the NLEEs, the appearance and the applications of the soliton theory. By the way, the properties of the soliton solutions are introduced in this chapter.In chapter two, the similarity reduction is studied, which reducts the high order or high dimension NLEEs. The similarity reduction makes it easy to sovle the NLEEs. The differences among the classical Lie Group method, nonclassical Lie Group method and the derect CK method, which belong to similarity reduction are shown. At the end of this chapter, the five-order dispersive equation is inducted to ordinary differencial equation.M. L. Wang's homogeneous balance method is studied, through which can get lots of significative exact solutions in many physical fields, mean while the Backlund transformation, and can serch for similarity reduction and multi-soliton solutions, etc. At the end of this chapter, a new exact solutions of variable-coefficient Huxley equation is investigated.Chapter four in which Hirota method is studied is an important part of this paper. The Hirota method, developed in 70s, brings NLEEs to arithmetic operators, which can predigest the complex NLEEs. Through this method, we can get Backlund transformation, lax pair, etc. This paper studies the development of Hirota method, and a new one-soliton solution and multi-soliton solution are investigated.The Painleve analysis is discussed, which tests the genetating of the NLEEs, the singularity and Painleve property. Painleve ODE and PDE Test are introduced also. The ODE from five order dispersive equation is tested by Painleve ODE Test here.