The Study On Exact Solutions Of Partial Differential Equations And Interaction In The Solitary Waves

This paper is based on the theory of linear and nonlinear partial differential equations and technic of computer symbolic computation. The works we have done include: First, using Painleve singularity structure analysis method, we have proved that the coupled Schrodinger -KdV equations admit Painleve property; Second, according to the truncated Painleve expansion technique, rational transformation method and "degree" method, we obtained the Hirota bilinear form of the coupled Schrodinger-KdV equations and the Caudrey-Dodd-Gibbon-Kaeada equation. Then, by means of the Hirota bilinear form, the single-soliton and two-soliton solutions of these equations are educed; Third, some new exact solutions of the Burgers-BBM equation, the Burgers-Fisher equation and the generalized Burgers-Fisher equation have been obtained by means of the first integral method which introduced by Feng ZhaoSheng in 2002; Forth, the interaction in the solitary-wave solutions of the integrable models are studied, and it has been found that for some integrable models, completely non-elastic interactions may occur when specific conditions between the wave vectors and velocities are satisfied. In this paper, we concentrate on a special phenomenon—fusion phenomenon about Chaffee-Infante (CI) equation.The study on the nonlinear phenomenon of natural science has procured important evolution followed quantum mechanics and theory of relativity in 20 century. Einstein ever pointed: because the basal equations on physics are nonlinear, all research of the mathematics and physics must start from beginning. In fact, up to now, nonlinear science has already developed to become one of the top tasks of many subjects. In 1978, Ablowitz and Segur found that all the ordinary differential equation (ODE) which obtained by similarity reduction from PDE that could be solved by inverse scattering technique possess Painleve property. so they give a conjecture—Painleve conjecture: "Every ordinary differential equation which arises as a similarity reduction of a completely integrable partial differential equations is of Painleve type, perhaps after a transformation of variables." This conjecture provides a necessary condition which can testify a PDE is whether or not completely integral. Then, Weiss, Tabor and Carnevale introduced the concept of Painleve property of PDE and Painleve PDE test, which made the estimation whether a PDE possesses Painleve property more convenient and concise.The coupled Schrodinger -KdV (CSK) system is a (1+1) dimensional nonlinear evolution equation. We have showed in second Chapter the CSK equation possesses the Painleve property. And in third Chapter, using the result we obtained in second Chapter, the one-soliton and two-soliton solutions have been gained for CSK equations by the Hirota bilinear method. Furthermore, the Hirota bilinear form of the CDGK equation is gained by means of three methods: rational transformation method,"degree"method and the truncated Painleve expansion technique. Then, utilizing the Hirota bilinear method, we have get the one-soliton and two-soliton solutions of CDGK equation.In 2002, Feng ZhaoSheng introduced the new method—the first integral method—to solve nonlinear partial differential equations effectively. The main idea of this method can be expressed as follows: for a nonlinear partial differential equation, assuming it has a traveling wave solution, so this PDE can be changed to an ODE. Then making a transformation on the ODE, and let it become a set of one order equations. The excellent idea of the first integral method is using the Division Theorem to seek one first integral of the set of one order equations, which can reduce the ODE to a first-order integrable ordinary differential equation, and then we can obtain the exact solutions of PDE by the direct integral method. By means of the first integral method, we have studied the Burgers-BBM equation, the Burgers-Fisher equation and the generalized Burgers-Fisher equation. Some new exact solutions of these equations are obtained. It does not seem that these new results have been presented previously.At last, in fifth Chapter we discussed the problem of the interaction in the solitary-wave solutions of integrable nonlinear partial differential equations. Usually, the interaction in the solitary-wave solutions of nonlinear partial differential equations can be obtained by the methods such as IST or Hirota bilinear method to be considered completely elastic. However, it has been found that for some integrable models, completely non-elastic interactions may occur when specific conditions between the wave vectors and velocities are satisfied. For instance, at a specific time, one soliton may fission to two or more solitons; or on the contrary, two or more solitons will fusion to one soliton. Those two types of phenomena are called as soliton fission and soliton fusion respectively. In this paper, we concentrate on fusion phenomenon about Chaffee-Infante (CI) equation.