映射法求解非线性系统模型解析解的研究
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摘要
波动是自然界中最常见的现象之一,波动问题遍及物理学、数学、力学、光学、化学、生物学、地理学、通信工程、机械工程、航空和航天技术等自然科学和技术的各个领域。随着科学技术水平的提高,人们发现自然科学和工程技术各个领域中普遍存在各自的非线性现象。许多非线性现象可用非线性系统的数学模型——非线性波动方程(即:非线性演化方程)来描述。人们通过求解这些非线性演化方程,寻找其解析解来揭示出各种非线性现象的奥秘和内在规律,为人类认识自然和改造自然服务。本学位论文的研究内容是改进和发展各种映射法,把它们用于各类非线性偏微分方程和随机偏微分方程的研究中,找出更多新的解析解。主要的工作如下:
改进了椭圆方程映射法,并被我们用于(1+1)-维变系数Sawada-Kotere方程的研究中,获得了许多新的Jacobian椭圆函数多项式的解和双曲函数多项式的解,包括了单种Jacobian椭圆函数的多项式形式的解,单种双曲函数的多项式形式的解,两种Jacobian椭圆函数混合的多项式形式的解,两种双曲函数混合的多项式形式的解,不仅有实函数表达式解,而且有复函数表达式解。改进了变系数投影Riccati方程映射法,并被用于(2+1)-维简化的广义的Broer-Kaup系统研究中,文献(Huang DJ, Zhang HQ. Chaos, Solitons & Fractals2005;23:601)的方法仅是我们方法的一个特例,我们的方法不仅能获得被Huang方法得到的所有解析解,而且还有许多新的解析解。广义的Broer-Kaup系统解的范围也从实数集扩张到复数集。
借助Exp-函数法,找到了广义Riccati方程的一个有理指数函数解,这个有理指数函数解包含现有各种Riccati方程映射法中给出的所有类型Riccati方程的各种类型的三角函数解和双曲函数解。为此,我们提出由广义Riccati方程和它的一个有理指数函数解和一个有理数解为基础,构造出一种新的Riccati方程映射法--有理指数映射法,它能把现有的各种Riccati方程映射法和tanh函数法完美地统一起来。我们把它用于求解耦合的mKdV方程的研究中,得到的耦合的mKdV方程有理指数解析解,如果给定这个有理指数解析解的各参数值,就能迅速写出耦合的mKdV方程所具有的各类三角函数和双曲函数解析解的具体形式。
在上述的研究基础,一种双Riccati方程的有理指数映射法被提出,且分别被用于耦合的(1+1)-维Whitham-Broer-Kaup方程和(2+1)-维Broer-Kaup-Kupershmidt方程的研究中,分别得到了这两个方程的新的有理指数组合解的形式。
提出把映射法拓展到非线性随机偏微分方程的研究领域,我们把Riccati方程映射法用于Wick型广义随机Korteweg-de Vries方程和Wick型随机mKdV方程的研究,借助厄米变换和白噪声理论,在白噪声条件下,得到了这个两个方程的双曲函数-指数函数型、三角函数-指数函数型和指数函数型三种类型的解析解。
改进了变系数投影Riccati方程映射方法,推广到Wick型随机偏微分方程研究中。把该方法用于(2+1)-维Wick型广义随机Broer-Kaup系统的研究中,在白噪声环境下,获得了丰富的Wick型广义随机Broer-Kaup系统的解析解,不仅得到了该方程的实函数型解析解,而且得到了它的复函数型解析解。
椭圆方程映射法也被应用于非线性随机偏微分方程研究领域。用椭圆方程映射法研究了两种不同类型的Wick型随机Korteweg-de Vries方程,获得了这两个随机方程的大量椭圆函数多项式的解析解,其中包括大量实函数型解析解和复函数型解析解。我们将Jacobian函数展开法看作一种特殊映射法,用于Wick型随机Korteweg-de Vries方程的研究,获得用椭圆方程映射法无法得到一些解析解,Jacobian函数展开法是对椭圆方程映射法的一个很好补充。
本论文的创新点主要有:
借助广义的Riccati方程的有理指数函数解,一种新的Riccati方程映射法--有理指数映射法被提出。有理指数映射法很好地把Riccati方程映射法和tanh函数法完美地统一起来。有理指数映射法能得到各种Riccati方程映射方法和tanh函数方法能够得到的所有结果,而且包含更多新的结果。
改进了变系数投影Riccati方程映射法和椭圆方程映射法,在Riccati方程有理指数映射法基础上,提出了双Riccati方程的有理指数映射法,这些方法被用于多个非线性偏微分方程的研究中,获得了许多新的结果。
将Riccati方程映射法、变系数投影Riccati方程映射法、椭圆方程映射法引入到Wick型非线性随机物理模型的研究中。在白噪声条件下,获得丰富的不同类型的精确的解析解,其中许多解析解是其他方法是无法得到的。
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.
改进了椭圆方程映射法,并被我们用于(1+1)-维变系数Sawada-Kotere方程的研究中,获得了许多新的Jacobian椭圆函数多项式的解和双曲函数多项式的解,包括了单种Jacobian椭圆函数的多项式形式的解,单种双曲函数的多项式形式的解,两种Jacobian椭圆函数混合的多项式形式的解,两种双曲函数混合的多项式形式的解,不仅有实函数表达式解,而且有复函数表达式解。改进了变系数投影Riccati方程映射法,并被用于(2+1)-维简化的广义的Broer-Kaup系统研究中,文献(Huang DJ, Zhang HQ. Chaos, Solitons & Fractals2005;23:601)的方法仅是我们方法的一个特例,我们的方法不仅能获得被Huang方法得到的所有解析解,而且还有许多新的解析解。广义的Broer-Kaup系统解的范围也从实数集扩张到复数集。
借助Exp-函数法,找到了广义Riccati方程的一个有理指数函数解,这个有理指数函数解包含现有各种Riccati方程映射法中给出的所有类型Riccati方程的各种类型的三角函数解和双曲函数解。为此,我们提出由广义Riccati方程和它的一个有理指数函数解和一个有理数解为基础,构造出一种新的Riccati方程映射法--有理指数映射法,它能把现有的各种Riccati方程映射法和tanh函数法完美地统一起来。我们把它用于求解耦合的mKdV方程的研究中,得到的耦合的mKdV方程有理指数解析解,如果给定这个有理指数解析解的各参数值,就能迅速写出耦合的mKdV方程所具有的各类三角函数和双曲函数解析解的具体形式。
在上述的研究基础,一种双Riccati方程的有理指数映射法被提出,且分别被用于耦合的(1+1)-维Whitham-Broer-Kaup方程和(2+1)-维Broer-Kaup-Kupershmidt方程的研究中,分别得到了这两个方程的新的有理指数组合解的形式。
提出把映射法拓展到非线性随机偏微分方程的研究领域,我们把Riccati方程映射法用于Wick型广义随机Korteweg-de Vries方程和Wick型随机mKdV方程的研究,借助厄米变换和白噪声理论,在白噪声条件下,得到了这个两个方程的双曲函数-指数函数型、三角函数-指数函数型和指数函数型三种类型的解析解。
改进了变系数投影Riccati方程映射方法,推广到Wick型随机偏微分方程研究中。把该方法用于(2+1)-维Wick型广义随机Broer-Kaup系统的研究中,在白噪声环境下,获得了丰富的Wick型广义随机Broer-Kaup系统的解析解,不仅得到了该方程的实函数型解析解,而且得到了它的复函数型解析解。
椭圆方程映射法也被应用于非线性随机偏微分方程研究领域。用椭圆方程映射法研究了两种不同类型的Wick型随机Korteweg-de Vries方程,获得了这两个随机方程的大量椭圆函数多项式的解析解,其中包括大量实函数型解析解和复函数型解析解。我们将Jacobian函数展开法看作一种特殊映射法,用于Wick型随机Korteweg-de Vries方程的研究,获得用椭圆方程映射法无法得到一些解析解,Jacobian函数展开法是对椭圆方程映射法的一个很好补充。
本论文的创新点主要有:
借助广义的Riccati方程的有理指数函数解,一种新的Riccati方程映射法--有理指数映射法被提出。有理指数映射法很好地把Riccati方程映射法和tanh函数法完美地统一起来。有理指数映射法能得到各种Riccati方程映射方法和tanh函数方法能够得到的所有结果,而且包含更多新的结果。
改进了变系数投影Riccati方程映射法和椭圆方程映射法,在Riccati方程有理指数映射法基础上,提出了双Riccati方程的有理指数映射法,这些方法被用于多个非线性偏微分方程的研究中,获得了许多新的结果。
将Riccati方程映射法、变系数投影Riccati方程映射法、椭圆方程映射法引入到Wick型非线性随机物理模型的研究中。在白噪声条件下,获得丰富的不同类型的精确的解析解,其中许多解析解是其他方法是无法得到的。
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.
引文
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