非线性偏微分方程的B（？）cklund变换
|
摘要
随着科学的发展,非线性现象出现在自然科学与工程技术等许多领域,对应的非线性模型也变得复杂多样,因此描述这些模型的非线性偏微分方程成为重要的研究课题.非线性偏微分方程有许多求解方法,Backlund变换法为其中一种,一方面它可以由方程的已知解导出另一个解,如果重复应用可求出此方程的多孤子解,另一方面它还可以由已知方程的解推出另一个方程的解.因此Backlund变换是求解偏微分方程行之有效的方法.
本文求出Burgers方程和uxxx=auux+but的可积系统,并推导出此可积系统下方程所有的(自)Backlund变换.文章中得到的结论和用通常的方法得出的结论有所不同,它给出了可积系统下方程所有的自Backlund变换.最后,由偏微分方程的可积系统导出了方程之间的Backlund变换(Miura变换).这些变换为求解非线性偏微分方程更多新的精确解奠定了基础.
本文安排如下：
第一章简单概述Backlund变换,并举例给出两个求方程自Backlund变换常用的方法：WTC方法与扩展齐次平衡法.
第二章求Burgers equation,uxxx=auux+but方程的可积系统,导出在此可积系统下方程所有的自Backlund变换.
第三章运用可积系统推出一类简单非线性方程之间的Miura变换.
With the development of science, non-linear phenomena appear in the natu-ral sciences, engineering technology and many other areas, then the correspond-ing non-linear models are complicated. Non-linear equations, describing the above models, become an important research topic. There are many methods to solve non-linear partial differential equations, one of which is Backlund trans-formation. On one hand, this approach could construct a new solution from the known solution, and obtain Multi-soliton solutions of the original equation by repeated application. On the other hand, the solution of another equation could be deduced from the one of known equation via the approach. Thus Backlund transformation is an effective method to solve partial differential equations.
In this paper, we find the integrable systems of Burgers equation and uχχχ= auuχ+but, then deduce all Auto-Backlund transformation of those equations under the above integrable systems. The conclusions obtained are dif-ferent from the ones under other methods, and they give all the Auto-Backlund transformation under the integrable system of equations. In the end, the Miura transformation between the equations is obtained by the integrable system of partial differential equations. These transformations are foundation for getting more new exact solutions of non-linear partial differential equations.
This article is arranged as follows:
Chapter 1 introduces the Backlund transformation, and gives two methods to find the Auto-Backlund transformation by some examples:WTC method and extended homogeneous balance method.
Chapter 2 finds the integrable systems of Burgers equation and uχχχ= auuχ+but, under which we derive all of the Auto-Backlund transformation of the equations.
Chapter 3 deduces a class of Miura transformations between simple non-linear equations by their integrable systems.
本文求出Burgers方程和uxxx=auux+but的可积系统,并推导出此可积系统下方程所有的(自)Backlund变换.文章中得到的结论和用通常的方法得出的结论有所不同,它给出了可积系统下方程所有的自Backlund变换.最后,由偏微分方程的可积系统导出了方程之间的Backlund变换(Miura变换).这些变换为求解非线性偏微分方程更多新的精确解奠定了基础.
本文安排如下：
第一章简单概述Backlund变换,并举例给出两个求方程自Backlund变换常用的方法：WTC方法与扩展齐次平衡法.
第二章求Burgers equation,uxxx=auux+but方程的可积系统,导出在此可积系统下方程所有的自Backlund变换.
第三章运用可积系统推出一类简单非线性方程之间的Miura变换.
With the development of science, non-linear phenomena appear in the natu-ral sciences, engineering technology and many other areas, then the correspond-ing non-linear models are complicated. Non-linear equations, describing the above models, become an important research topic. There are many methods to solve non-linear partial differential equations, one of which is Backlund trans-formation. On one hand, this approach could construct a new solution from the known solution, and obtain Multi-soliton solutions of the original equation by repeated application. On the other hand, the solution of another equation could be deduced from the one of known equation via the approach. Thus Backlund transformation is an effective method to solve partial differential equations.
In this paper, we find the integrable systems of Burgers equation and uχχχ= auuχ+but, then deduce all Auto-Backlund transformation of those equations under the above integrable systems. The conclusions obtained are dif-ferent from the ones under other methods, and they give all the Auto-Backlund transformation under the integrable system of equations. In the end, the Miura transformation between the equations is obtained by the integrable system of partial differential equations. These transformations are foundation for getting more new exact solutions of non-linear partial differential equations.
This article is arranged as follows:
Chapter 1 introduces the Backlund transformation, and gives two methods to find the Auto-Backlund transformation by some examples:WTC method and extended homogeneous balance method.
Chapter 2 finds the integrable systems of Burgers equation and uχχχ= auuχ+but, under which we derive all of the Auto-Backlund transformation of the equations.
Chapter 3 deduces a class of Miura transformations between simple non-linear equations by their integrable systems.
引文
[1]Wazwaz A..The extended tanh method for the Zakharov-Kuznetsov(ZK) equation,the modified ZK equation,and its generalized forms[J].Commun Nonlnear Sci Numer Simul,2008,13(6):1039-1047
[2].Wazwaz A.. New travelling wave solutions to the Boussinesq and the Klein-Gordon equations[J].Commun Nonlinear Sci Numer Simul.2008, 13(5):889-901
[3]Feng Z..The first-integral method to study the Burgers-Korteweg-de Vries equation[J].J.Phys.A.,2002,35:343-349
[4]Feng Z..On explicit exact solutions to the compound Burgers-KdV equa-tion[J].Phys.Lett.A.,2002,293:57-66
[5]Bekir A..Application of the((?))-expansion method for nonlinear evolution equations[J].Phys.Lett.A.,2008,372:3400-3406
[6]Wang M.,Li x.,Zhang J.,et al.The((?))-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Phys.Lett.A.,2008,372(4):417
[7]Lamb G..Elements of soliton theory[R].NewYork:Wiley,1980
[8]Chen Y..Chaos Solitons and Fractals.2003,17:693
[9]范恩贵.大连理工大学博士学位论文[D].1998
[10]谷超豪等著.孤立子理论及其应用[M].浙江科学技术出版社,1990
[11]陈陆君,梁昌洪著.孤立子理论及其应用[M].西安电子科技出版社,1997
[12]Rogers C., Shadwiek W.. Backlund transformations and their applica-tion[M].Aeademie Press,New York,1982
[13]Wahlquist H., Estabrook F.. Backlund transformations for solitons of Ko-rteweg-de-Vies equation[J]. Phys. Rev. Lett.,1973,31:1386-1389
[14]Weiss J., Carnvale G.. The painlev property for partial differential equa-tions[J]. J. Math. phys.,1983,24:522-526
[15]Weiss J.. The Painleve property and Backlund transformations for the se-quence of Boussinesq equations[J]. J. Math. Phys.,1985,26(2):258-269
[16]Darboux G.. Compts rendus hebdomadaires des Seanees de I academie des scicnccs[R]. pairs,1882,94:1456
[17]Seott J.. Report on waves[R], Fourteen meeting of the British assoeiation for the ad-vancement of selenee, John Murray, London,1544,311
[18]Matveev V., Cand M.. Darboux transformation and Solitons[R], Berlin: Springer,1991
[19]谷超豪等.孤立子理论中的Darboux变换及其几何应用[M].上海：上海科技出版社,1999
[20]Wadati M.. Simple derivation of Backlund transform Riccati form of inverse method[J]. prog. Theor. phys.,1975,53:418
[21]Hu X.. Lax pairs and Backlund transformation for a coupled Ramani equa-tion and its related system[J]. Appl. Math. Lett..2000,13:45
[22]Hu X. and Zhu Z.. A Backlund transformation and nonlinear superposition formula for the Belov-Chaltikian lattice[J]. J. phys. A.,1998,31:4755
[23]Gordoa P.. Obtaining Backlund transformations for higher order ordi-nary differential equations[J]. Real Sociedad Matematica Espanola,2003, 4:255-264
[24]Weiss J., Tabor M., Carnevale G., et al. The Painleve property for partial differential equations [J]. J. Math. Phys.,1983,24(3):522
[25]Weiss J.. The Painleve property and Backlund transformations for the se-quence of Boussinesq equa-tions[J]. J. Math. Phys.,1985,26(2):258-269
26] Wang Mingliang. Homogeneous balance method and applications[J]. Phys. Lett. A.,1993,213(2):279-284
[27]Wang Mingliang, Zhou Yubin, Li Zhibi. Application of a homogeneous bal-ance method to exact solutions of nonlinear equations in mathematical physics[J]. Phys. Lett. A.,1996,216(1):67-75
[28]Lamb G.. Backlund transformations for certain nonlinear evolution equa-tions[J], J. Math. Phys.,1974,15:2157-2165
[29]Rogers C. and Shadwick W.. Backlund transformations and their applica-tions[M]. Academic Press, New York,1982
[30]wahlquist H., Estabrook F.. Prolongation structures of nonlinear evolution equations[J]. J. Math. Phys.,1975,16:161-167
[31]Wu H.. On Backlund transformations for nonlinear partial differential equa-tions[J]. J. Math Anal. Appl.,1995,192:151-179
[32]Chern S., Tenenblat K.. Pseudospherical surfacesand evolutionequations[J]. Stud. Appl. Math.,1986,74:55-83
[33]Tian C.. Backlund transformation of nonlinear evolution equations[J]. Acta. Math. Appl.,1985,2:87-94
[34]Wu H.. On Backlund transformations for nonlinear partial differential equa-tions[J]. J. Math. Anal. Appl.,1995,192:151-179
[35]Wu H.. On Miura transformations among nonlinear partial differential equa-tions[J]. J. Math. Phys.,2006,47:1-19
[2].Wazwaz A.. New travelling wave solutions to the Boussinesq and the Klein-Gordon equations[J].Commun Nonlinear Sci Numer Simul.2008, 13(5):889-901
[3]Feng Z..The first-integral method to study the Burgers-Korteweg-de Vries equation[J].J.Phys.A.,2002,35:343-349
[4]Feng Z..On explicit exact solutions to the compound Burgers-KdV equa-tion[J].Phys.Lett.A.,2002,293:57-66
[5]Bekir A..Application of the((?))-expansion method for nonlinear evolution equations[J].Phys.Lett.A.,2008,372:3400-3406
[6]Wang M.,Li x.,Zhang J.,et al.The((?))-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Phys.Lett.A.,2008,372(4):417
[7]Lamb G..Elements of soliton theory[R].NewYork:Wiley,1980
[8]Chen Y..Chaos Solitons and Fractals.2003,17:693
[9]范恩贵.大连理工大学博士学位论文[D].1998
[10]谷超豪等著.孤立子理论及其应用[M].浙江科学技术出版社,1990
[11]陈陆君,梁昌洪著.孤立子理论及其应用[M].西安电子科技出版社,1997
[12]Rogers C., Shadwiek W.. Backlund transformations and their applica-tion[M].Aeademie Press,New York,1982
[13]Wahlquist H., Estabrook F.. Backlund transformations for solitons of Ko-rteweg-de-Vies equation[J]. Phys. Rev. Lett.,1973,31:1386-1389
[14]Weiss J., Carnvale G.. The painlev property for partial differential equa-tions[J]. J. Math. phys.,1983,24:522-526
[15]Weiss J.. The Painleve property and Backlund transformations for the se-quence of Boussinesq equations[J]. J. Math. Phys.,1985,26(2):258-269
[16]Darboux G.. Compts rendus hebdomadaires des Seanees de I academie des scicnccs[R]. pairs,1882,94:1456
[17]Seott J.. Report on waves[R], Fourteen meeting of the British assoeiation for the ad-vancement of selenee, John Murray, London,1544,311
[18]Matveev V., Cand M.. Darboux transformation and Solitons[R], Berlin: Springer,1991
[19]谷超豪等.孤立子理论中的Darboux变换及其几何应用[M].上海：上海科技出版社,1999
[20]Wadati M.. Simple derivation of Backlund transform Riccati form of inverse method[J]. prog. Theor. phys.,1975,53:418
[21]Hu X.. Lax pairs and Backlund transformation for a coupled Ramani equa-tion and its related system[J]. Appl. Math. Lett..2000,13:45
[22]Hu X. and Zhu Z.. A Backlund transformation and nonlinear superposition formula for the Belov-Chaltikian lattice[J]. J. phys. A.,1998,31:4755
[23]Gordoa P.. Obtaining Backlund transformations for higher order ordi-nary differential equations[J]. Real Sociedad Matematica Espanola,2003, 4:255-264
[24]Weiss J., Tabor M., Carnevale G., et al. The Painleve property for partial differential equations [J]. J. Math. Phys.,1983,24(3):522
[25]Weiss J.. The Painleve property and Backlund transformations for the se-quence of Boussinesq equa-tions[J]. J. Math. Phys.,1985,26(2):258-269
26] Wang Mingliang. Homogeneous balance method and applications[J]. Phys. Lett. A.,1993,213(2):279-284
[27]Wang Mingliang, Zhou Yubin, Li Zhibi. Application of a homogeneous bal-ance method to exact solutions of nonlinear equations in mathematical physics[J]. Phys. Lett. A.,1996,216(1):67-75
[28]Lamb G.. Backlund transformations for certain nonlinear evolution equa-tions[J], J. Math. Phys.,1974,15:2157-2165
[29]Rogers C. and Shadwick W.. Backlund transformations and their applica-tions[M]. Academic Press, New York,1982
[30]wahlquist H., Estabrook F.. Prolongation structures of nonlinear evolution equations[J]. J. Math. Phys.,1975,16:161-167
[31]Wu H.. On Backlund transformations for nonlinear partial differential equa-tions[J]. J. Math Anal. Appl.,1995,192:151-179
[32]Chern S., Tenenblat K.. Pseudospherical surfacesand evolutionequations[J]. Stud. Appl. Math.,1986,74:55-83
[33]Tian C.. Backlund transformation of nonlinear evolution equations[J]. Acta. Math. Appl.,1985,2:87-94
[34]Wu H.. On Backlund transformations for nonlinear partial differential equa-tions[J]. J. Math. Anal. Appl.,1995,192:151-179
[35]Wu H.. On Miura transformations among nonlinear partial differential equa-tions[J]. J. Math. Phys.,2006,47:1-19