Researches On Exact Solutions For The Model Of Nonlinear System Using The Mapping Method

The wave is one of pervasive phenomena in the nature and it exists in many scopesof science and technology, such as physics, mathematics, mechanics, optics, chemistry,biology, communication engineering, mechanism engineering, engineering of aviationand space?ight and so on. With the development of science and technology, an increas-ing number of researchers realize that nonlinear phenomena widely exist in the variousfields of science and technology. A number of nonlinear phenomenon can be describedby the mathematical models—-the nonlinear systems which are nonlinear wave equa-tions (i.e. nonlinear evolution equations). To serve realizing nature and altering nature,the researchers expose the arcanum and the law themselves of various nonlinear phe-nomenon by investigating the solutions of the nonlinear evolution equations. The mainwork of this doctoral dissertation is modifying and developing various mapping methodsand seeking more new solutions of nonlinear partial di?erential equations and nonlinearstochastic partial di?erential equations. The main contents of this dissertation include:Modifying elliptic equation mapping method and applying it to the investigationof (1+1)-dimensional Sawada-Kotere equation with variable coe?cient, we obtain manysolutions with the polynomial of Jacobian functions and with the polynomial of hyper-bolic functions, which include the polynomials comprised by single Jacobian function,or by double Jacobian functions, or by single hyperbolic function, or by double hyper-bolic functions. They contain not only the solutions of real function but also complexfunction.We have improved variable-coe?cient projective Riccati equation mappingmethod and applied it to the research of (2+1)-dimensional simplified generalized Broer-Kaup system, and found Huang's method (Huang DJ, Zhang HQ. Chaos, Solitons &Fractals 2005;23:601) is only a special case of our method. We can derive not only allsolutions previously obtained by Huang's method, but also many new solutions. Therange of the solution is also expanded from real number field to complex number filed.With the help of Exp-function method, we deduce a rational-exponent solution toa generalized Riccati equation. The rational-exponent solution include all solutions oftrigonometric function and hyperbolic function for various Riccati equations. Using thegeneralized Riccati equation and its rational-exponent solution along with its rational so-lution, we construct a new Riccati equation method which is called rational-exponentmapping method. the new method can unify all kinds of Riccati equation mappingmethod and tanh-function method ideally. The rational-exponent mapping method isused to investigate the coupling mKdV equation and many rational-exponent solutionsare derived by us. If all parameters of rational-exponent solution are given, we canrapidly write various solutions of trigonometric function and hyperbolic function of thecoupling mKdV equation.Based on above results, the multiple Riccati equation rational-exponent mapping method is proposed and applied to the coupling (1+1)-dimensional Whitham-Broer-Kaup equation and the (2+1)-dimensional Broer-Kaup-Kupershmidt system. The com-bined solutions of rational-exponent function and rational function of above equationsare obtained.We propose to expand Riccati equation mapping method to the researching fieldof nonlinear stochastic partial di?erential equation and apply it to Wick-type general-ized stochastic Korteweg-de Vries equation and Wick-type generalized stochastic mKdVequation. With Hermite transformation and white noise theory, we deduce three kinds ofsolutions of hyperbolic-exponential type, trigonometric-exponential type and exponen-tial type for the Wick-type generalized stochastic Korteweg-de Vries equation and theWick-type generalized stochastic mKdV equation in white noise.A modified variable-coe?cient projective Riccati equation method is extendedfrom nonlinear partial di?erential equation to nonlinear stochastic partial di?erentialequation. Applying it to (2+1)-dimensional Wick-type generalized stochastic Broer-Kaup system, we get abundant solutions to Wick-type generalized stochastic Broer-Kaupsystem in white noise. They have not only real function solutions but also complex func-tion solutions.Elliptic equation mapping method is used to investigate nonlinear stochastic par-tial di?erential equation. Solving two kinds of Wick-type stochastic Korteweg-de Vriesequations with the elliptic equation mapping method, we derive many Jacobian func-tions solutions to the two stochastic Korteweg-de Vries equations, which include manyreal function solutions and complex function solutions. Taking Jacobian elliptic functionexpansion method as a special case of elliptic equation mapping method and applying itto a stochastic Korteweg-de Vries equation, we obtain many new Jacobian elliptic func-tion solutions which can not be deduced by elliptic equation mapping method. It meansthat Jacobian elliptic function expansion method is complementarity for elliptic equationmapping method.The innovations of this dissertation are as follows:By the aid of a rational-exponent solution and a rational-function solution for ageneralized Riccati equation, a generalized Riccati equation method which is calledrational-exponent mapping method is constructed. the new mapping method can unifyall kinds of Riccati equation mapping method and tanh-function method ideally. Therational-exponent mapping method can obtain not only all solutions derived by variousRiccati equation mapping method and tanh-function method, but also more new results.We improved variable-coe?cient projective Riccati equation method and modifiedelliptic equation mapping method. Based on The rational-exponent mapping method,a multiple Riccati equation rational-exponent mapping method is also proposed. Thesemethods are applied to many nonlinear partial di?erential equation and many new resultsare obtained.Riccati equation method, variable-coe?cient projective Riccati equation method, elliptic equation mapping method, Jacobian elliptic function expansion method etc. areextended from the field of nonlinear partial di?erential equation to nonlinear stochasticpartial di?erential equation. In white noise, we obtain abundant solutions to many Wick-type stochastic partial di?erential equations while some of them can not be derived byother method.