非线性偏微分方程的精确求解
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摘要
孤立子理论是非线性科学的一个重要组成部分,许多理论和应用科学中的数学模型导出的非线性方程具有孤立子特性。因此,孤立子方程的求解在理论和应用中都具有极其重要的意义。本文根据数学机械化思想,以符号计算软件为工具,研究了非线性发展方程的求解问题,利用符号计算系统Maple,并应用改进的求解方法解得一些方程的新的精确解。
第一章是绪论,介绍了孤立子研究的历史和发展的概况,包括孤立子理论的起源,非线性发展方程的求解方法的发展过程,还介绍了数学机械化和符号计算的概念和应用。
第二章内容是应用F-展开法求解广义Hirota-Satsuma耦合方程。首先介绍F-展开法的步骤,接着应用F-展开法来求解Hirota-Satsuma耦合方程,得到了很多文献[14]中没有给出的新的方程的精确解。
第三章内容是利用文献[13]提出的改进的耦合的Riccati方程组来求解(2+1)维Burgers方程。首先介绍了利用改进的耦合的Riccati方程组来求解的方法步骤,接着应用此方法,得到了(2+1)维Burgers方程的更多的新的精确解。
第四章首先利用改进的Riccati方程求得KdV-MKdV方程的一些新解,其次求解2+1维广义浅水波方程的类孤子解和周期解,求得的解带有变系数,由于系数的可变性,可获得更多的方程的类孤子解和周期解.
第五章首先简要的介绍了求解非线性发展方程的一种有效的方法——达布变换法,其次提出了一种新的达布变换,并加以证明,最后用新的达布变换对于BK系统进行应用,求得了方程的新的解.
The soliton theory is an important part of the nonlinear science. There are many nonlinear partial differential equations that have solution properties in the pure and applied science. Therefore, solving soliton Equations is very important in the theory and in the application. This dissertation, under the guidance of mathematics mechanization and by means of symbolic computation software, considers the exact solutions to the nonlinear partial differential equations. With the help of symbolic computation system Maple, by the application of improved methods of solving equations, some new exact solutions are presented.
Chapter1 is to introduce the history and development of the soliton theory, including the origin of the soliton theory, development of the method solving nonlinear equations, also introduced mathematical mechanization and symbolic computation and application.
Chapter2 is the application of the F-expansion method to the generalized Hirota -Satsuma coupled equations. First on the F-expansion method steps, and then by the F-expansion method , we get a lot of the new exacted solutions that can not be find in reference .
Chapter 3 is to introduce the solving of (2+1)-dimensional Burgers equation by the improved coupling Riccati equations raised by reference [13]. First of all, we describe the steps of the F-expansion method , and then apply the method, many of the new exact solutions of (2+1)-dimensional Burgers equation are obtained.
Chapter 4 is to firstly introduce the solving of KdV-MKdV equation by the improved Riccati equation . Secondly, we obtained the like soliton solutions and the periodic solutions of the (2+1)-dimensional generalized shallow-water wave equation which are with variable coefficient, because of the variable coefficient ,the equation will be have more like soliton solutions and periodic solutions.
Chapter 5 begins with a brief introduction to solving the nonlinear development equation by an effective method - Darboux transformation method, followed by a new Darboux transformation is presented with the proof, finally, the new Darboux transformation is applied to BK system, then we obtained the new solutions of BK system.
第一章是绪论,介绍了孤立子研究的历史和发展的概况,包括孤立子理论的起源,非线性发展方程的求解方法的发展过程,还介绍了数学机械化和符号计算的概念和应用。
第二章内容是应用F-展开法求解广义Hirota-Satsuma耦合方程。首先介绍F-展开法的步骤,接着应用F-展开法来求解Hirota-Satsuma耦合方程,得到了很多文献[14]中没有给出的新的方程的精确解。
第三章内容是利用文献[13]提出的改进的耦合的Riccati方程组来求解(2+1)维Burgers方程。首先介绍了利用改进的耦合的Riccati方程组来求解的方法步骤,接着应用此方法,得到了(2+1)维Burgers方程的更多的新的精确解。
第四章首先利用改进的Riccati方程求得KdV-MKdV方程的一些新解,其次求解2+1维广义浅水波方程的类孤子解和周期解,求得的解带有变系数,由于系数的可变性,可获得更多的方程的类孤子解和周期解.
第五章首先简要的介绍了求解非线性发展方程的一种有效的方法——达布变换法,其次提出了一种新的达布变换,并加以证明,最后用新的达布变换对于BK系统进行应用,求得了方程的新的解.
The soliton theory is an important part of the nonlinear science. There are many nonlinear partial differential equations that have solution properties in the pure and applied science. Therefore, solving soliton Equations is very important in the theory and in the application. This dissertation, under the guidance of mathematics mechanization and by means of symbolic computation software, considers the exact solutions to the nonlinear partial differential equations. With the help of symbolic computation system Maple, by the application of improved methods of solving equations, some new exact solutions are presented.
Chapter1 is to introduce the history and development of the soliton theory, including the origin of the soliton theory, development of the method solving nonlinear equations, also introduced mathematical mechanization and symbolic computation and application.
Chapter2 is the application of the F-expansion method to the generalized Hirota -Satsuma coupled equations. First on the F-expansion method steps, and then by the F-expansion method , we get a lot of the new exacted solutions that can not be find in reference .
Chapter 3 is to introduce the solving of (2+1)-dimensional Burgers equation by the improved coupling Riccati equations raised by reference [13]. First of all, we describe the steps of the F-expansion method , and then apply the method, many of the new exact solutions of (2+1)-dimensional Burgers equation are obtained.
Chapter 4 is to firstly introduce the solving of KdV-MKdV equation by the improved Riccati equation . Secondly, we obtained the like soliton solutions and the periodic solutions of the (2+1)-dimensional generalized shallow-water wave equation which are with variable coefficient, because of the variable coefficient ,the equation will be have more like soliton solutions and periodic solutions.
Chapter 5 begins with a brief introduction to solving the nonlinear development equation by an effective method - Darboux transformation method, followed by a new Darboux transformation is presented with the proof, finally, the new Darboux transformation is applied to BK system, then we obtained the new solutions of BK system.
引文
[1]张玉峰,张鸿庆.组合KdV与MKdV方程Backlund变换及其一类精确解. 2001,41(4) 392-395.
[2] Hirota R. The direct method in soliton theory. Cambridge: Cambridge Univer- sity Press; 2004.
[3] Chen Y, Li B.General projective Riccati equation method and exact solutions for generalized Kdv-type and Kdv-Burgers-type equations with nonlinear terms of any order. Chaos,Solitons and Fractals 2004;19:977.
[4] Gu C H et al. Darboux transformation in soliton theory and its geometric applications. Shanghai:Shanghai Scientiˉc and Technical Publishers; 1999.
[5] Rogers C, Schief WK.B?acklund and Darboux transformation, geometry and modern applications in soliton theory. Cambridge: Cambridge University Press; 2002.
[6] Matveev VB, Salle MA.Darboux transformation and solitons. Berlin:Springer; 1991.
[7] Yan Z Y, Zhang H Q. New Auto-Darboux Transformation and Explicit Analytic Solutions for the Generalized KdV Equation with External Force Term. Journal of Mathematical Research Exposition 21(2001)323-328.
[8] Fan E G. Solving Kadomtsev-Petviashvili Equation via a New Decomposition and Darboux Transformation. Commun. Theor. Phys. 37 (2002) 145-148.
[9] Li X M, Chen A H. Darboux transformation and multi-soliton solutions of Boussinesq-Burgers equation. Physics Letters A 342(2005) 413-420.
[10] Wu T Y, Zhang J E, Cook L P,Roythurd V and Tulin M (Eds)1996 SLAM Math is for Solving Problems, 233-241.
[11] Li Y S, Ma W X, Zhang J E. Darboux transformations of classical Boussinesq system and its new solutions. Physics Letters A 275 (2000) 60-66.
[12] Li Y S, Zhang J E. Darboux transformations of classical Boussinesq system and its multi-soliton solutions. Physics Letters A 284 (2001)253-258.
[13] Fuding Xie,Xiaoshan Gao. Exact traveling wave solutions for a class of nonlinear partial differential equations. Chaos,Solitons and Fractals 19(2004)1113-1117.
[14]范恩贵.可积系统与计算机代数.2004.北京.科学出版社.
[15] Liu Chunping et al .A simple method to construct the traveling wave solutions to nonlinear evolution equations .Physics Letter A.1998,246 : 113~116
[16] En-Gui Fan traveling wave solutions for nonlinear equations using symbolic computation.computers&mathematics.43(2002)671-680
[17]张辉群.齐次平衡方法的扩展及应用.数学物理学报. 2001.21A(3):321-325
[18]张金良,任东锋,王明亮,方宗德.Davey-Stewartson I的周期波解.数学物理学报.2005,25A(2):213-219
[19] Zhang Jin-liang,Zhang Ling-yuan,Wang Ming-liang.Periodic Wave Solutions forKonopelchenko-Dubrovsky Equation. Chin.Quart.J.of Math.2005,20(1):72-78
[20]梅建琴,张鸿庆.2+1维广义浅水波方程的类孤子解与周期解.数学物理学报.2005,25A(6):784-788
[21] Yong Chen, Biao Li. General projective Riccati Equation method and exact solutions for generalized KdV-type and KdV-Burgers-type equations with nonlinear terms of any order. Chaos, Solitons and Fractals 19(2004)977-984
[22] Figen Kangalgil, Fatma Ayaz.New exact travelling wave solutions for the Ostrovsky equation. Physics Letters A.327(2008)1831-1835.
[23]范恩贵,张鸿庆.齐次平衡法若干新的应用.数学物理学报.1999,19(3) :286-292
[24]周振江,李志斌.Broer-Kaup系统的达布变换和新的精确解.acta physica sinica.52(02),2003:263-265
[25]郑莹.非线性发展方程精确求解中若干问题的研究: (博士学位论文).大连:大连理工大学,2007.
[26]智红燕.非线性偏微分方程求解和对称约化: (博士学位论文).大连:大连理工大学,2007.
[27]张渊渊.孤子方程求解中的若干构造性技巧: (博士学位论文).大连:大连理工大学,2007.
[28] Qi Wang,Yong Chen,Zhang Hongqing.A new Jacobi elliptic function rational expansion method and its application to (1+1)-dimensional dispersive long wave equation. Chaos, Solitons and Fractals.23(2005)477-483.
[29] New exact solutions to a system of coupled KdV equations.Engui Fan,Hongqing Zhang.Physics Letters A 245 (1998) 389-392.
[30]陈勇.孤立子理论中的若干问题的研究及机械化实现:(博士学位论文).大连:大连理工大学,2003.
[31]王明亮,李志斌,周宇斌.齐次平衡原则及其应用.兰州大学学报(自然科学版)0455-2059(1999)03-0008-09.
[32] Ablowitz MJ,Clarkson PA. Solitons, nonlinear evolution equations and inverse scatting. Cambridge: Cambridge University Press:1991.
[2] Hirota R. The direct method in soliton theory. Cambridge: Cambridge Univer- sity Press; 2004.
[3] Chen Y, Li B.General projective Riccati equation method and exact solutions for generalized Kdv-type and Kdv-Burgers-type equations with nonlinear terms of any order. Chaos,Solitons and Fractals 2004;19:977.
[4] Gu C H et al. Darboux transformation in soliton theory and its geometric applications. Shanghai:Shanghai Scientiˉc and Technical Publishers; 1999.
[5] Rogers C, Schief WK.B?acklund and Darboux transformation, geometry and modern applications in soliton theory. Cambridge: Cambridge University Press; 2002.
[6] Matveev VB, Salle MA.Darboux transformation and solitons. Berlin:Springer; 1991.
[7] Yan Z Y, Zhang H Q. New Auto-Darboux Transformation and Explicit Analytic Solutions for the Generalized KdV Equation with External Force Term. Journal of Mathematical Research Exposition 21(2001)323-328.
[8] Fan E G. Solving Kadomtsev-Petviashvili Equation via a New Decomposition and Darboux Transformation. Commun. Theor. Phys. 37 (2002) 145-148.
[9] Li X M, Chen A H. Darboux transformation and multi-soliton solutions of Boussinesq-Burgers equation. Physics Letters A 342(2005) 413-420.
[10] Wu T Y, Zhang J E, Cook L P,Roythurd V and Tulin M (Eds)1996 SLAM Math is for Solving Problems, 233-241.
[11] Li Y S, Ma W X, Zhang J E. Darboux transformations of classical Boussinesq system and its new solutions. Physics Letters A 275 (2000) 60-66.
[12] Li Y S, Zhang J E. Darboux transformations of classical Boussinesq system and its multi-soliton solutions. Physics Letters A 284 (2001)253-258.
[13] Fuding Xie,Xiaoshan Gao. Exact traveling wave solutions for a class of nonlinear partial differential equations. Chaos,Solitons and Fractals 19(2004)1113-1117.
[14]范恩贵.可积系统与计算机代数.2004.北京.科学出版社.
[15] Liu Chunping et al .A simple method to construct the traveling wave solutions to nonlinear evolution equations .Physics Letter A.1998,246 : 113~116
[16] En-Gui Fan traveling wave solutions for nonlinear equations using symbolic computation.computers&mathematics.43(2002)671-680
[17]张辉群.齐次平衡方法的扩展及应用.数学物理学报. 2001.21A(3):321-325
[18]张金良,任东锋,王明亮,方宗德.Davey-Stewartson I的周期波解.数学物理学报.2005,25A(2):213-219
[19] Zhang Jin-liang,Zhang Ling-yuan,Wang Ming-liang.Periodic Wave Solutions forKonopelchenko-Dubrovsky Equation. Chin.Quart.J.of Math.2005,20(1):72-78
[20]梅建琴,张鸿庆.2+1维广义浅水波方程的类孤子解与周期解.数学物理学报.2005,25A(6):784-788
[21] Yong Chen, Biao Li. General projective Riccati Equation method and exact solutions for generalized KdV-type and KdV-Burgers-type equations with nonlinear terms of any order. Chaos, Solitons and Fractals 19(2004)977-984
[22] Figen Kangalgil, Fatma Ayaz.New exact travelling wave solutions for the Ostrovsky equation. Physics Letters A.327(2008)1831-1835.
[23]范恩贵,张鸿庆.齐次平衡法若干新的应用.数学物理学报.1999,19(3) :286-292
[24]周振江,李志斌.Broer-Kaup系统的达布变换和新的精确解.acta physica sinica.52(02),2003:263-265
[25]郑莹.非线性发展方程精确求解中若干问题的研究: (博士学位论文).大连:大连理工大学,2007.
[26]智红燕.非线性偏微分方程求解和对称约化: (博士学位论文).大连:大连理工大学,2007.
[27]张渊渊.孤子方程求解中的若干构造性技巧: (博士学位论文).大连:大连理工大学,2007.
[28] Qi Wang,Yong Chen,Zhang Hongqing.A new Jacobi elliptic function rational expansion method and its application to (1+1)-dimensional dispersive long wave equation. Chaos, Solitons and Fractals.23(2005)477-483.
[29] New exact solutions to a system of coupled KdV equations.Engui Fan,Hongqing Zhang.Physics Letters A 245 (1998) 389-392.
[30]陈勇.孤立子理论中的若干问题的研究及机械化实现:(博士学位论文).大连:大连理工大学,2003.
[31]王明亮,李志斌,周宇斌.齐次平衡原则及其应用.兰州大学学报(自然科学版)0455-2059(1999)03-0008-09.
[32] Ablowitz MJ,Clarkson PA. Solitons, nonlinear evolution equations and inverse scatting. Cambridge: Cambridge University Press:1991.