两类非线性波动方程的行波解
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摘要
在科学应用中出现的重大问题中,各种各样非线性色散方程显式精确解的研究已经引起了人们的关注.这些显式精确解的研究,采用了数学上各种分析方法,诸如反散射法,达布变换法,双线性法,李群方法,动力系统分支理论,正余弦函数方法,双曲正切函数法,Fan函数展开法,齐次平衡方法等,目前,没有统一的方法可以用来处理所有类型的非线性微分方程.在寻找非线性扩散方程精确解的各种方法中,双曲正切函数法和正余弦函数方法是求精确解最直接而有效的代数方法之一.本文将在第三章采用扩展双曲正切函数法得到(N+1)维sine-cosine-Gordon方程的显式精确扭结波解(或反扭结波解)和显式精确周期波解.本文将在第四章采用A.M.Wazwaz推广的正余弦函数方法来研究一类非线性四阶广义的Camassa-Holm方程.从而显示这种方法是一种求解非线性微分方程的有效方法.最后对本文的工作进行了总结,提出了有待于进一步解决的问题.
Studies of various explicit exact solutions of nonlinear dispersive equations hadattracted much attention in connection with the important problems that arise in scientificapplications. Mathematically, these explicit exact solutions have been studied by usingvarious analytical methods, such as inverse scattering method, Darboux transformationmethod, Hirota bilinear method, Lie group method, bifurcation method of dynamic systems,sine-cosine method, tanh function method, Fan-expansion method, homogenous balancemethod and so on. Practically, there is no unified technique that can be employed to handleall types of nonlinear differential equations. The tanh method and the sine-cosine methodare ones of most direct and effective algebraic method for finding exact solutions ofnonlinear diffusion equations. In Chapter 3, the (N+1)-dimensional sine-cosine-Gordonequations are studied. The existence of kink (or anti-kink) traveling wave explicit exactsolutions and periodic traveling wave explicit exact solutions are proved, by using theextended tanh method. In Chapter 4, a class of nonlinear fourth order variant of ageneralized Camassa-Holm equation is studied by using the sine-cosine method extendedby A. M. Wazwaz. It is shown that the extended tanh method provides a powerfulmathematical tool for solving a great many nonlinear partial differential equations inmathematical physics. Finally, the summary of this thesis and the prospect of future studyare given.
Studies of various explicit exact solutions of nonlinear dispersive equations hadattracted much attention in connection with the important problems that arise in scientificapplications. Mathematically, these explicit exact solutions have been studied by usingvarious analytical methods, such as inverse scattering method, Darboux transformationmethod, Hirota bilinear method, Lie group method, bifurcation method of dynamic systems,sine-cosine method, tanh function method, Fan-expansion method, homogenous balancemethod and so on. Practically, there is no unified technique that can be employed to handleall types of nonlinear differential equations. The tanh method and the sine-cosine methodare ones of most direct and effective algebraic method for finding exact solutions ofnonlinear diffusion equations. In Chapter 3, the (N+1)-dimensional sine-cosine-Gordonequations are studied. The existence of kink (or anti-kink) traveling wave explicit exactsolutions and periodic traveling wave explicit exact solutions are proved, by using theextended tanh method. In Chapter 4, a class of nonlinear fourth order variant of ageneralized Camassa-Holm equation is studied by using the sine-cosine method extendedby A. M. Wazwaz. It is shown that the extended tanh method provides a powerfulmathematical tool for solving a great many nonlinear partial differential equations inmathematical physics. Finally, the summary of this thesis and the prospect of future studyare given.
引文
[1]陈守洙,江之水.普通物理学（第三册）[M].高等教育出版社, 2001: 68.
[2]谢鸿政,杨枫林.数学物理方程[M].科学出版社, 2005: 8-9.
[3]朱长江,邓引斌.偏微分方程教程[M].科学出版社, 2005:4.
[4]郭硕鸿.电动力学（第二版）[M].高等教育出版社, 2005: 135-137.
[5]高尚华.数学分析（第三版）（下册）[M].高等教育出版社, 2002: 157.
[6] J. Boussinesq. Théorie de l'intumescence liquide, applelée onde solitaire ou detranslation, se propageant dans un canal rectangulaire [J]. Comptes Rendus del'Academie des Sciences, 72: 755–759.
[7] J.Boussinesq. Théorie des ondes et des remous qui se propagent le long d'un canalrectangulaire horizontal, en communiquant au liquide contenu dans ce canal desvitesses sensiblement pareilles de la surface au fond [J]. Journal de MathématiquePures et Appliquées, Deuxième Sériex, 17: 55–108.
[8] Korteweg D J, de Vries G. On the change of rorm of form of long waves advancing ina rectangular canal and on a new type of long stationary waves [J]. Philos. Mag. Ser. ,1895, 39(5): 422-44.
[9] E. Schr?dinger. An Undulatory Theory of the Mechanics of Atoms and Molecules [J].Physical Review 28 (6): 1049–1070.
[10]M. J. Ablowitz, P. A. Clarkson, Solitons. Nonlinear Evolution Equations and InverseScattering [J]. Cambridge University Press, London, 1991.
[11] Gardner C S, Greene J M, Kruskal M D, et al.Method for solving the Kortweg-deVries equation[J]. Phys. Rev. Lett. , 1967, 19: 1095-1097.
[12]Lax P D. Integrals of nonlinear equations of and solitary waves [J]. Commun.PureAppl. Math. , 1968, 21: 467-490.
[13]Zakharov V E，Shabat A B. Exact theory of two-dimensional self-focusing and one-dimensional waves in nonlinear media[J]. Sov. Phys. JETP, 1972, 34: 62-69.
[14]Ablowitz M J, Kaup D J, Newell A C, et al. Method for solving the sine-Gordonequation[J]. Phys. Rev. Lett., 1973, 30: 1262-1264.
[15]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear evolution equations of physicalsignificance[J]. Phys. Rev. Lett. , 1974, 31: 125-127.
[16]Ablowitz M J, Kaup D J, Newell A C, et al. The inverse scattering transform-Fourieranalysis for nonlinear problems [J]. Study. in Appl. Math. , 1974, 53: 249-315.
[17]V. B. Matveev, M. A. Salle. Darboux Transformation and Solitons [J]. Springer-Verlag,Berlin, 1991.
[18]C. H. Gu, H. S. Hu, Z. X. Zhou. Darboux Transformationsin Soliton Theory anditsGeometric Applications, ShanghaiSci. Tech. Publ [J], Shanghai, 1999.
[19]R. Hirota, J. Satsuma. Soliton solutions of a coupled KdV equation [J]. Phys. Lett. A,85(1981): 407–408.
[20]Hirota R. Exact solutions of the Kortweg-de Vries equation for multiple colli-sions ofsolitons [J].
[21]Hirota R, Satsuma.Nonlinear evolution equations generated from the Backlund-transformation for the Boussinesq equation [J]. Prog. Theor. Phys, 1977, 57: 797-807.
[22]Hu X B. Hirota type equations, soliton solutions, Backlund transformations and conver-sation laws [J]. Partial Diff. Eqs. , 1990, 3: 87-95.
[23]Hu X B, Li Y. Nonlinear superposition formulate of Ito equation and a model equationfor shallow water waves [J]. Phys. A, 1991, 24: 1979-1986.
[24]Wang M L. Exact solutions for a compound KdV-Burgers equation [J]. Phys. Lett. A,1996, 213: 279-287.
[25]P. J. Olver. Applications of Lie Groupsto Differential Equations, Springer-Verlag [J].New York, 1993.
[26]G. W. Bluman, S. Kumei. Symmetries and Differential Equations. Springer-Verlag [J],Berlin, 1989.
[27]Y. Zheng, S. Lai, Peakons. Solitary patterns and periodicsolutionsfor generalizedCamassa–Holm equations [J], Phys. Lett. A, 372(2008): 4141–4143.
[28]J. B. Li，Z. Liu. Travelling wavesolutionsfora class ofnonlinear dispersive equations[J].Chin.Ann.Math, 23B(2002): 397–418.
[29]Shengqiang Tang, Wentao Huang. Bifurcations of travelling wave solutions for thegeneralized double sinh-Gordon equation [J]. Appl. Math. Comput, 189(2007):1774–1781.
[30]A. M. Wazwaz. A class of nonlinear fourth order variant of a generalizedCamassa–Holm equation with compactand noncompact solutions. Appl. Math.Comput [J], 165(2005): 485–501.
[31]L. Tian, J. Yin. New compacton solutionsand solitary wave solutions of fully nonlineargeneralized Camassa–Holm equations [J]. Chaos Solitons Fract, 20(2004): 289–299.
[32]A. M. Wazwaz. Solutions of compactand noncompact structures for nonlinear Klein–Gordon-type equation [J]. Appl. Math. Comput, 134(2003): 487–500.
[33]A. M. Wazwaz.A sine–cosine method for handling nonlinear wave equations [J]. Math.Comput. Model, 40(2004): 499–508.
[34]A. M. Wazwaz. Solitons and periodic solutions for the fifth-order KdV equation [J],Appl. Math. Lett, 19(2006): 1162–1167.
[35]A. M. Wazwaz. Analyticstudy on nonlinear variant of the RLW and the PHI-fourequations, Commun. Nonliear Sci.Nuer.Simul [J], 12(2007): 314–327.
[36]Z. Y. Yan, H. Q. Zhang. New explicitand exacttravelling wave solutions for a systemof variant Boussinesq equationsin mathematical physics [J]. Phys. Lett. A, 252(1999):291–296.
[37]A. M. Wazwaz. A reliabletreatment of the physicalstructure for the nonlinear equationK( m, n) [J]. Appl. Math. Comput, 163(2005): 1081–1095.
[38]Z. Y. Yan. New explicit travelling wave solutions for two new integrable coupled non-linear evolution equations [J]. Phys. Lett. A. 292(2001): 100–106.
[39]Malfliet W. Solitary wave solutions of nonlinear wave equations [J]. Am J Phys, 1992,60: 650-654.
[40]E. G. Fan. Uniformly constructing a series of explicit exact solutions to nonlinearequationsin mathematical physics [J]. Chaos Solitons Fract, 16(2003): 819–839.
[41]S. Zhang. New exact solutions of the KdV–Burgers–Kuramoto equation [J], Phys. Lett.A, 358(2006): 414–420.
[42]M. L. Wang. Exact solutions for a compound KdV–Burgersequation [J]. Phys. Lett. A,213 (1996): 279–287.
[43]Wang M L. Solitary wave solutions for variant Boussinesq equations [J]. Phys. Lett. A,1995, 199: 169-172.
[44]Wang M L, Zhou Y B, Li Z B. Applications of homogeneous balance method to exactsolutions of nonlinear evolution equations in mathematical physics [J]. Phys. Lett. A,1996, 216: 67-73.
[45]Fu Z T, Liu S K, Liu S D. New Jacobi elliptic function expansion and newperiodicwave solutions of nonlinear wave equations [J]. Phys. Lett. A, 2001, 290: 72-76.
[46]Fu Z T, Liu S K, Liu S D. New rational form solutions to mKdV equation [J].Commun. Theor. Phys. , 2005, 43: 423-426.
[47]Zhang J F, Guo G P, Wu F M. New multi-soliton solutions and traveling wavesolutions of disperse long-wave equations [J]. Chin. Phys. , 2002, 11(6): 533-536.
[48]Zhang J L, Wang Y M, Wang M L. New applications of the homogeneous balanceprinciple [J]. Chin. Phys. , 2003, 12(3): 245-250.
[49]李志斌.非线性数学物理方程的行波解[M].科学出版社, 2007：2.
[50]王彬炜.扩展双曲函数法与非线性方程的精确解[D].广州大学硕士学位论文,2007.
[51]C T Yan. A simple transformation for nonlinear waves [J]. Phys. Lett. A, 1996，224:77.
[52]郑赟,张鸿庆.一个非线性方程的显式行波解[J].物理学报, 2000,3:1.
[53]Yan Zhen-Ya, Zhang Hong-qing, Fan En-Gui. New Explicit and Travelling WaveSolutions for a Classs of Nonlinear Evolution Equations [J]. Acta Physica Sinica,48:1.
[54]Deng-Shan Wang, Zhenya Yan and Hongbo Li. Some special types of solutions of aclass of the (N + 1)-dimensional nonlinear wave equations [J], Comp. Math. Appl. 56(2008):1569-1579.
[55]Shengqiang Tang, Chunhai Li, Kelei Zhang. Bifurcations of traveling wave solutionsin the (N+1)-dimensional sine-cosine-Gorden equations [J]. Commun Nonlinear SciNumer Simulat, 2010: 3358-3366.
[56]P. A. Clarkson, T. J. Priestley, Symmetries of a class of nonlinear fourth order partialdifferential equations [J]. Nonlinear Math. Phys. 6 (1) (1999): 66-98.
[2]谢鸿政,杨枫林.数学物理方程[M].科学出版社, 2005: 8-9.
[3]朱长江,邓引斌.偏微分方程教程[M].科学出版社, 2005:4.
[4]郭硕鸿.电动力学（第二版）[M].高等教育出版社, 2005: 135-137.
[5]高尚华.数学分析（第三版）（下册）[M].高等教育出版社, 2002: 157.
[6] J. Boussinesq. Théorie de l'intumescence liquide, applelée onde solitaire ou detranslation, se propageant dans un canal rectangulaire [J]. Comptes Rendus del'Academie des Sciences, 72: 755–759.
[7] J.Boussinesq. Théorie des ondes et des remous qui se propagent le long d'un canalrectangulaire horizontal, en communiquant au liquide contenu dans ce canal desvitesses sensiblement pareilles de la surface au fond [J]. Journal de MathématiquePures et Appliquées, Deuxième Sériex, 17: 55–108.
[8] Korteweg D J, de Vries G. On the change of rorm of form of long waves advancing ina rectangular canal and on a new type of long stationary waves [J]. Philos. Mag. Ser. ,1895, 39(5): 422-44.
[9] E. Schr?dinger. An Undulatory Theory of the Mechanics of Atoms and Molecules [J].Physical Review 28 (6): 1049–1070.
[10]M. J. Ablowitz, P. A. Clarkson, Solitons. Nonlinear Evolution Equations and InverseScattering [J]. Cambridge University Press, London, 1991.
[11] Gardner C S, Greene J M, Kruskal M D, et al.Method for solving the Kortweg-deVries equation[J]. Phys. Rev. Lett. , 1967, 19: 1095-1097.
[12]Lax P D. Integrals of nonlinear equations of and solitary waves [J]. Commun.PureAppl. Math. , 1968, 21: 467-490.
[13]Zakharov V E，Shabat A B. Exact theory of two-dimensional self-focusing and one-dimensional waves in nonlinear media[J]. Sov. Phys. JETP, 1972, 34: 62-69.
[14]Ablowitz M J, Kaup D J, Newell A C, et al. Method for solving the sine-Gordonequation[J]. Phys. Rev. Lett., 1973, 30: 1262-1264.
[15]Ablowitz M J, Kaup D J, Newell A C, et al. Nonlinear evolution equations of physicalsignificance[J]. Phys. Rev. Lett. , 1974, 31: 125-127.
[16]Ablowitz M J, Kaup D J, Newell A C, et al. The inverse scattering transform-Fourieranalysis for nonlinear problems [J]. Study. in Appl. Math. , 1974, 53: 249-315.
[17]V. B. Matveev, M. A. Salle. Darboux Transformation and Solitons [J]. Springer-Verlag,Berlin, 1991.
[18]C. H. Gu, H. S. Hu, Z. X. Zhou. Darboux Transformationsin Soliton Theory anditsGeometric Applications, ShanghaiSci. Tech. Publ [J], Shanghai, 1999.
[19]R. Hirota, J. Satsuma. Soliton solutions of a coupled KdV equation [J]. Phys. Lett. A,85(1981): 407–408.
[20]Hirota R. Exact solutions of the Kortweg-de Vries equation for multiple colli-sions ofsolitons [J].
[21]Hirota R, Satsuma.Nonlinear evolution equations generated from the Backlund-transformation for the Boussinesq equation [J]. Prog. Theor. Phys, 1977, 57: 797-807.
[22]Hu X B. Hirota type equations, soliton solutions, Backlund transformations and conver-sation laws [J]. Partial Diff. Eqs. , 1990, 3: 87-95.
[23]Hu X B, Li Y. Nonlinear superposition formulate of Ito equation and a model equationfor shallow water waves [J]. Phys. A, 1991, 24: 1979-1986.
[24]Wang M L. Exact solutions for a compound KdV-Burgers equation [J]. Phys. Lett. A,1996, 213: 279-287.
[25]P. J. Olver. Applications of Lie Groupsto Differential Equations, Springer-Verlag [J].New York, 1993.
[26]G. W. Bluman, S. Kumei. Symmetries and Differential Equations. Springer-Verlag [J],Berlin, 1989.
[27]Y. Zheng, S. Lai, Peakons. Solitary patterns and periodicsolutionsfor generalizedCamassa–Holm equations [J], Phys. Lett. A, 372(2008): 4141–4143.
[28]J. B. Li，Z. Liu. Travelling wavesolutionsfora class ofnonlinear dispersive equations[J].Chin.Ann.Math, 23B(2002): 397–418.
[29]Shengqiang Tang, Wentao Huang. Bifurcations of travelling wave solutions for thegeneralized double sinh-Gordon equation [J]. Appl. Math. Comput, 189(2007):1774–1781.
[30]A. M. Wazwaz. A class of nonlinear fourth order variant of a generalizedCamassa–Holm equation with compactand noncompact solutions. Appl. Math.Comput [J], 165(2005): 485–501.
[31]L. Tian, J. Yin. New compacton solutionsand solitary wave solutions of fully nonlineargeneralized Camassa–Holm equations [J]. Chaos Solitons Fract, 20(2004): 289–299.
[32]A. M. Wazwaz. Solutions of compactand noncompact structures for nonlinear Klein–Gordon-type equation [J]. Appl. Math. Comput, 134(2003): 487–500.
[33]A. M. Wazwaz.A sine–cosine method for handling nonlinear wave equations [J]. Math.Comput. Model, 40(2004): 499–508.
[34]A. M. Wazwaz. Solitons and periodic solutions for the fifth-order KdV equation [J],Appl. Math. Lett, 19(2006): 1162–1167.
[35]A. M. Wazwaz. Analyticstudy on nonlinear variant of the RLW and the PHI-fourequations, Commun. Nonliear Sci.Nuer.Simul [J], 12(2007): 314–327.
[36]Z. Y. Yan, H. Q. Zhang. New explicitand exacttravelling wave solutions for a systemof variant Boussinesq equationsin mathematical physics [J]. Phys. Lett. A, 252(1999):291–296.
[37]A. M. Wazwaz. A reliabletreatment of the physicalstructure for the nonlinear equationK( m, n) [J]. Appl. Math. Comput, 163(2005): 1081–1095.
[38]Z. Y. Yan. New explicit travelling wave solutions for two new integrable coupled non-linear evolution equations [J]. Phys. Lett. A. 292(2001): 100–106.
[39]Malfliet W. Solitary wave solutions of nonlinear wave equations [J]. Am J Phys, 1992,60: 650-654.
[40]E. G. Fan. Uniformly constructing a series of explicit exact solutions to nonlinearequationsin mathematical physics [J]. Chaos Solitons Fract, 16(2003): 819–839.
[41]S. Zhang. New exact solutions of the KdV–Burgers–Kuramoto equation [J], Phys. Lett.A, 358(2006): 414–420.
[42]M. L. Wang. Exact solutions for a compound KdV–Burgersequation [J]. Phys. Lett. A,213 (1996): 279–287.
[43]Wang M L. Solitary wave solutions for variant Boussinesq equations [J]. Phys. Lett. A,1995, 199: 169-172.
[44]Wang M L, Zhou Y B, Li Z B. Applications of homogeneous balance method to exactsolutions of nonlinear evolution equations in mathematical physics [J]. Phys. Lett. A,1996, 216: 67-73.
[45]Fu Z T, Liu S K, Liu S D. New Jacobi elliptic function expansion and newperiodicwave solutions of nonlinear wave equations [J]. Phys. Lett. A, 2001, 290: 72-76.
[46]Fu Z T, Liu S K, Liu S D. New rational form solutions to mKdV equation [J].Commun. Theor. Phys. , 2005, 43: 423-426.
[47]Zhang J F, Guo G P, Wu F M. New multi-soliton solutions and traveling wavesolutions of disperse long-wave equations [J]. Chin. Phys. , 2002, 11(6): 533-536.
[48]Zhang J L, Wang Y M, Wang M L. New applications of the homogeneous balanceprinciple [J]. Chin. Phys. , 2003, 12(3): 245-250.
[49]李志斌.非线性数学物理方程的行波解[M].科学出版社, 2007：2.
[50]王彬炜.扩展双曲函数法与非线性方程的精确解[D].广州大学硕士学位论文,2007.
[51]C T Yan. A simple transformation for nonlinear waves [J]. Phys. Lett. A, 1996，224:77.
[52]郑赟,张鸿庆.一个非线性方程的显式行波解[J].物理学报, 2000,3:1.
[53]Yan Zhen-Ya, Zhang Hong-qing, Fan En-Gui. New Explicit and Travelling WaveSolutions for a Classs of Nonlinear Evolution Equations [J]. Acta Physica Sinica,48:1.
[54]Deng-Shan Wang, Zhenya Yan and Hongbo Li. Some special types of solutions of aclass of the (N + 1)-dimensional nonlinear wave equations [J], Comp. Math. Appl. 56(2008):1569-1579.
[55]Shengqiang Tang, Chunhai Li, Kelei Zhang. Bifurcations of traveling wave solutionsin the (N+1)-dimensional sine-cosine-Gorden equations [J]. Commun Nonlinear SciNumer Simulat, 2010: 3358-3366.
[56]P. A. Clarkson, T. J. Priestley, Symmetries of a class of nonlinear fourth order partialdifferential equations [J]. Nonlinear Math. Phys. 6 (1) (1999): 66-98.