Levi方程的Painleve分析和精确解
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摘要
非线性微分方程的可积性与求解是非线性科学中一个重要的研究课题.而Painleve分析方法是判定其可积性和求解的一个有力工具.本文针对两个高阶Levi方程,做了以下工作:
(1).对(1+1)维Levi方程,利用WTC方法对其进行Painleve测试,仿照Painleve求解常微分方程(组)调谐因子的方法得到(1+1)维Levi方程的调谐因子,并通过Painleve测试证明了方程具有Painleve性质,在Painleve意义下可积(即P-可积).然后通过相容性分析得出了相容方程,从而将方程组的解的奇异流形展式截断为有限项的形式,利用Schwarz导数及其性质得到了方程的一类精确解.同时得到了方程的一个自Backlund变换.
(2).对(2+1)维Levi方程,利用WTC方法对其进行Painleve测试,对特殊情况下的耦合(2+1)维Levi方程进行分析求解,得到了方程的单孤子解,双孤子解及多孤子解.
(3).对于求得的Levi方程的精确解,选择适当的参数,利用Matlab作出了解的图形,并根据图形分析解的性质.在对耦合(2+1)维Levi方程双孤子解图像的分析中,讨论了孤立波的裂变现象.
(4).对(1+1)维Levi方程主导项系数的其他情况进行介绍,并进行Painleve测试,得到了这些情况下的相容方程组.
The integrable property for nonlinear differencial equations is important in nonlinear science. In this thesis, two higher order Levi equation are studied :
(1). We make use of WTC method to the (1+1)-dimensional Levi equation. It is proved that the equation possess Painleve property (P-integrable). The resonances are obtained by the WTC method. The compatible equations are obtained and the expansions of solution are truncated through the analysis of compatibility. Thereby, the exact solutions and Daboux-Backlund Transformation of the Levi equation are obtained by means of Schwarzian derivative.
(2). As to the (2+1)-dimensional Levi equation, we make use of WTC method to do the Painleve Test. Furthermore, One-soliton solution, Two-soliton solution, and N-soliton solution are obtained in some particular conditions.
(3). We use Matlab to get graphics of Levi equation, and do some analysis about the solutions. Furthermore, the fission phenomenon in Two-soliton solution of the coupled (2+1)-dimensional Levi equation is discussed.
(4). We introduce some special condition of the (1+1)-dimensional Levi equation in Painleve test, The compatible equations are obtained.
(1).对(1+1)维Levi方程,利用WTC方法对其进行Painleve测试,仿照Painleve求解常微分方程(组)调谐因子的方法得到(1+1)维Levi方程的调谐因子,并通过Painleve测试证明了方程具有Painleve性质,在Painleve意义下可积(即P-可积).然后通过相容性分析得出了相容方程,从而将方程组的解的奇异流形展式截断为有限项的形式,利用Schwarz导数及其性质得到了方程的一类精确解.同时得到了方程的一个自Backlund变换.
(2).对(2+1)维Levi方程,利用WTC方法对其进行Painleve测试,对特殊情况下的耦合(2+1)维Levi方程进行分析求解,得到了方程的单孤子解,双孤子解及多孤子解.
(3).对于求得的Levi方程的精确解,选择适当的参数,利用Matlab作出了解的图形,并根据图形分析解的性质.在对耦合(2+1)维Levi方程双孤子解图像的分析中,讨论了孤立波的裂变现象.
(4).对(1+1)维Levi方程主导项系数的其他情况进行介绍,并进行Painleve测试,得到了这些情况下的相容方程组.
The integrable property for nonlinear differencial equations is important in nonlinear science. In this thesis, two higher order Levi equation are studied :
(1). We make use of WTC method to the (1+1)-dimensional Levi equation. It is proved that the equation possess Painleve property (P-integrable). The resonances are obtained by the WTC method. The compatible equations are obtained and the expansions of solution are truncated through the analysis of compatibility. Thereby, the exact solutions and Daboux-Backlund Transformation of the Levi equation are obtained by means of Schwarzian derivative.
(2). As to the (2+1)-dimensional Levi equation, we make use of WTC method to do the Painleve Test. Furthermore, One-soliton solution, Two-soliton solution, and N-soliton solution are obtained in some particular conditions.
(3). We use Matlab to get graphics of Levi equation, and do some analysis about the solutions. Furthermore, the fission phenomenon in Two-soliton solution of the coupled (2+1)-dimensional Levi equation is discussed.
(4). We introduce some special condition of the (1+1)-dimensional Levi equation in Painleve test, The compatible equations are obtained.
引文
[1] M. J. Ablowitz and H. Segur. Exact linearization of a Painleve transcendent[J], Phys. Rev. Lett., 1977, 38: 1103-1106.
[2] M. J. Ablowitz, A. Ramani and H. Segur. Nonlinear evolutions and ordinary differencial equations of Painleve type[J], Lett. Nuovo Cim., 1978, 23: 333-338.
[3] P. Deift and E. Trubowitz. Inverse scattering on the line[J], Comm. Pure. Appl. Math., 1979, 32: 121-251.
[4] J. S. Russell. "Report on waves", Report of the 14 th meeting of British Association for the Advancement of Science[R], John Murray, London, 1844, 311-390.
[5] D. J. Korteweg and G. de Vries. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[J], Phil. Mag., 1895, 39: 422-443.
[6] A. Fermi, J. Pasta and S. Ulam. Studies of nonlinear proboems[R], Los Alamos Report, LA 1940, 1955: Lectures in Applied Mathematics, A. C. Newell ed., American Mathematical society, 1974, 15: 143-196.
[7] M. D. Kruskal and N. J. Zabusky. Progress on the Fermi-Pasta-Ulam nonlinear string problem[R], Princeton Plasma Physics Laboratory Annual Rept, MATT-Q-21, Primceton, N. J. pp., 1963, 301-308.
[8] N. J. Zabusky and M. D. Kruskal. Interaction of solitons in a collisionless plasma and the recurrence of initial states[J], Phys. Rev. Lett. , 1965, 15: 240-243.
[9] J. Weiss. Backlund transformation Lax Pairs and the Schwarzian derivative[J], J. Math. Phys., 24(6), 1983, 1405.
[10] Virgil Pierce. Painleve Analysis and Integrability[R], Report Submitted to Prof. A. Goriely for RTG, Spring, 1999.
[11]楼森岳,唐晓艳.非线性数学物理方法[M],科学出版社, 2006.
[12] J. D. Gibbon, P. Radmore, M. Tabor and D. Wood. The Painleve property and Hirota's method[J], Stud. Appl. Math., 1985, 72: 39-63.
[13] T. Alagesan and K. Porsezian. Singularity Structure Aanlysis and Hirota'sBilinearisation of the Coupled Integrable Dispersionless Equations[J], Chaos Solitons and Fractals, 1997, 8(7): 1645-1650.
[14] P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves[J], Comm. Pure Appl. Math., 968, 21: 467-490.
[15] M. J. Ablowitz, P. A. Clarkson. Soliton. Nonlinear Evolution Equations and Inverse Scattering[R], New York: Cambridge University Press, 1991.
[16]王明亮.非线性发展方程与孤立子[M],兰州:兰州大学出版社, 1990.
[17] J. Weiss, M. Tabor and G. Garnevale. Painleve Property for Partial differential equations[J]. , J. Math. Phys., 1983, 24: 522-526.
[18] J. Weiss. The Painleve property for partial differential equation[J], J. Math Phys., 1983, 24(3):522-526.
[19] A. C. Newell, M. Tabor and Y. B. Zeng. A unified approach to Painleve expansions, Physica[J], 1987, 29: 1-68.
[20]朱佐农.若干非线性偏微分方程的Painleve性质和Backlund变换[J],东南大学学报, 1994年第24卷第6期.
[21]叶彩儿.几个非线性发展方程(组)的精确解与Painleve分析[D],浙江大学硕士学位论文, 2003.
[22] A. Roy. Chowdhury. Painleve analysis and its applications[J], Boca Raton: CRC Press, 2000.
[23]张玉峰,代美丽. (2+1)维Levi族及其扩展可积模型[J],洛阳大学学报, 2006年第21卷第2期.
[24] Levi D. Sym A and Rauch-Wojciechowski[J], Phys. A: Math. Gen., 1983, 16: 2423.
[25] Yongtang Wu and Jinshun Zhang. Quasi-periodic solution of a new (2+1)- dimensional coupled soliton equation[J], Phys. A: Math. Gen., 2001, 34: 193-210.
[26]张金顺,李华夏. 2+1维Levi孤子方程的Darboux变换[J],郑州大学学报, 2001年9月第33卷第3期.
[27] Yufeng Zhang, Xiurong Guo. A(2+1) dimensional integrable hierarchy and its extending integrable model[J], Chaos Solitons and Fractals, 2006, 27: 555-559.
[28]谷超豪.孤立子理论与应用[M],杭州:浙江科技出版社, 1990.
[29]陈登远. Backlund变换与n孤子解[J],数学研究与评论, 2005年8月第25卷第3期.
[30] J. Weiss. The Painleve property for partial differential equations I: Backlund Transformation, Lax Pair, and the Schwarizan derivative[J], J. Math. Phys., 1983, 24(6): 1405-1413.
[2] M. J. Ablowitz, A. Ramani and H. Segur. Nonlinear evolutions and ordinary differencial equations of Painleve type[J], Lett. Nuovo Cim., 1978, 23: 333-338.
[3] P. Deift and E. Trubowitz. Inverse scattering on the line[J], Comm. Pure. Appl. Math., 1979, 32: 121-251.
[4] J. S. Russell. "Report on waves", Report of the 14 th meeting of British Association for the Advancement of Science[R], John Murray, London, 1844, 311-390.
[5] D. J. Korteweg and G. de Vries. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[J], Phil. Mag., 1895, 39: 422-443.
[6] A. Fermi, J. Pasta and S. Ulam. Studies of nonlinear proboems[R], Los Alamos Report, LA 1940, 1955: Lectures in Applied Mathematics, A. C. Newell ed., American Mathematical society, 1974, 15: 143-196.
[7] M. D. Kruskal and N. J. Zabusky. Progress on the Fermi-Pasta-Ulam nonlinear string problem[R], Princeton Plasma Physics Laboratory Annual Rept, MATT-Q-21, Primceton, N. J. pp., 1963, 301-308.
[8] N. J. Zabusky and M. D. Kruskal. Interaction of solitons in a collisionless plasma and the recurrence of initial states[J], Phys. Rev. Lett. , 1965, 15: 240-243.
[9] J. Weiss. Backlund transformation Lax Pairs and the Schwarzian derivative[J], J. Math. Phys., 24(6), 1983, 1405.
[10] Virgil Pierce. Painleve Analysis and Integrability[R], Report Submitted to Prof. A. Goriely for RTG, Spring, 1999.
[11]楼森岳,唐晓艳.非线性数学物理方法[M],科学出版社, 2006.
[12] J. D. Gibbon, P. Radmore, M. Tabor and D. Wood. The Painleve property and Hirota's method[J], Stud. Appl. Math., 1985, 72: 39-63.
[13] T. Alagesan and K. Porsezian. Singularity Structure Aanlysis and Hirota'sBilinearisation of the Coupled Integrable Dispersionless Equations[J], Chaos Solitons and Fractals, 1997, 8(7): 1645-1650.
[14] P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves[J], Comm. Pure Appl. Math., 968, 21: 467-490.
[15] M. J. Ablowitz, P. A. Clarkson. Soliton. Nonlinear Evolution Equations and Inverse Scattering[R], New York: Cambridge University Press, 1991.
[16]王明亮.非线性发展方程与孤立子[M],兰州:兰州大学出版社, 1990.
[17] J. Weiss, M. Tabor and G. Garnevale. Painleve Property for Partial differential equations[J]. , J. Math. Phys., 1983, 24: 522-526.
[18] J. Weiss. The Painleve property for partial differential equation[J], J. Math Phys., 1983, 24(3):522-526.
[19] A. C. Newell, M. Tabor and Y. B. Zeng. A unified approach to Painleve expansions, Physica[J], 1987, 29: 1-68.
[20]朱佐农.若干非线性偏微分方程的Painleve性质和Backlund变换[J],东南大学学报, 1994年第24卷第6期.
[21]叶彩儿.几个非线性发展方程(组)的精确解与Painleve分析[D],浙江大学硕士学位论文, 2003.
[22] A. Roy. Chowdhury. Painleve analysis and its applications[J], Boca Raton: CRC Press, 2000.
[23]张玉峰,代美丽. (2+1)维Levi族及其扩展可积模型[J],洛阳大学学报, 2006年第21卷第2期.
[24] Levi D. Sym A and Rauch-Wojciechowski[J], Phys. A: Math. Gen., 1983, 16: 2423.
[25] Yongtang Wu and Jinshun Zhang. Quasi-periodic solution of a new (2+1)- dimensional coupled soliton equation[J], Phys. A: Math. Gen., 2001, 34: 193-210.
[26]张金顺,李华夏. 2+1维Levi孤子方程的Darboux变换[J],郑州大学学报, 2001年9月第33卷第3期.
[27] Yufeng Zhang, Xiurong Guo. A(2+1) dimensional integrable hierarchy and its extending integrable model[J], Chaos Solitons and Fractals, 2006, 27: 555-559.
[28]谷超豪.孤立子理论与应用[M],杭州:浙江科技出版社, 1990.
[29]陈登远. Backlund变换与n孤子解[J],数学研究与评论, 2005年8月第25卷第3期.
[30] J. Weiss. The Painleve property for partial differential equations I: Backlund Transformation, Lax Pair, and the Schwarizan derivative[J], J. Math. Phys., 1983, 24(6): 1405-1413.