活动标架法在Whitham-Broer-Kaup系统中的应用（英文）
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摘要
正规化Maurer-Cartan形式的基是寻找Whitham-Broer-Kaup系统的解的不变性的重要工具,由于Whitham-Broer-Kaup系统的非线性和经典活动标架法的局限性,该系统的正规化Maurer-Cartan形式的基尚未被给出。基于等变活动标架理论,运用Maple软件,给出了求解Whitham-Broer-Kaup系统正规化Maurer-Cartan形式的基的一种有效方法。提供的方法克服了经典活动标架法的局限性,只用到无穷小决定方程组和截面正规化的选取,甚至没有用到活动标架、微分不变量、不变微分算子的显式表达式,是一种非常高效的算法。结果可用于研究Whitham-Broer-Kaup系统的解的不变性,并将有助于进一步研究海洋、大气、非线性动力学等领域中运动的规律和趋势。
The basis of the invariantized Maurer-Cartan forms of Whitham-Broer-Kaup system is the important tool to find the solutions' invariant properties of Whitham-Broer-Kaup system.Because of the nonlinear complicacy and limitations of classic moving frames,the basis of the invariantized Maurer-Cartan forms of this system has not been given as far as we know.In this paper,based on the equivariant moving frames theory,with the Maple software,an efficient method is provided to obtain a basis of the invariantized Maurer-Cartan forms of Whitham-BroerKaup system.This method overcomes the limitations of classic moving frames and is very efficient,only using the infinitesimal determining equations and choice of cross-section normalization to completely reveal the complete basis of the invariantized Maurer-Cartan forms,without any explicit formulas for either the moving frame or differential invariants and invariant differential operators.These conclusions in this paper can be used to find the solutions' invariant properties of WhithamBroer-Kaup system and further study the laws and trends of motion in oceanic science,atmospheric science,nonlinear dynamics,etc.
The basis of the invariantized Maurer-Cartan forms of Whitham-Broer-Kaup system is the important tool to find the solutions' invariant properties of Whitham-Broer-Kaup system.Because of the nonlinear complicacy and limitations of classic moving frames,the basis of the invariantized Maurer-Cartan forms of this system has not been given as far as we know.In this paper,based on the equivariant moving frames theory,with the Maple software,an efficient method is provided to obtain a basis of the invariantized Maurer-Cartan forms of Whitham-BroerKaup system.This method overcomes the limitations of classic moving frames and is very efficient,only using the infinitesimal determining equations and choice of cross-section normalization to completely reveal the complete basis of the invariantized Maurer-Cartan forms,without any explicit formulas for either the moving frame or differential invariants and invariant differential operators.These conclusions in this paper can be used to find the solutions' invariant properties of WhithamBroer-Kaup system and further study the laws and trends of motion in oceanic science,atmospheric science,nonlinear dynamics,etc.
引文
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[6]XIE Fuding,YAN Zhenya,ZHANG Hongqing.Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations[J].Phys Lett A,2001,285:76-80.
[7]FAN Engui,ZHANG Hongqing.A new approach to Bcklund transformations of nonlinear evolution equations[J].Appl Math Mech,1998,19:645-650.
[8]LISLE I G,REID G J.Cartan structure of infinite Lie pseudo-groups[C]∥Geometrical Approaches to Differential Equations,PJVassiliou and IG Lisle eds,2000:116-145.
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[12]OLVER P J,POHJANPELTO J.Maurer-Cartan forms and the structure of Lie pseudo-groups[J].Selecta Math,2005,11:99-126.
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[17]KOGAN I A,OLVER P J.Invariants of objects and their images under surjective maps[J].Lobachevskii J Math,2015,36:260-285.
[18]OLVER P J.The symmetry groupoid and weighted signature of a geometric object[J].J Lie Theory,2015,26:235-267.
[19]OLVER P J.Recursive moving frames[J].Results Math,2011,60:423-452.
[20]OLVER P J.Differential invariant algebras[J].Contemp Math,2011,549:95-121.
[21]ROBERTT,FRANCIS V.Group foliation of finite difference equations[J].CommunNonlinear Sci,2018,59:235-254.
[2]BROER L J F.Approximate equations for long water waves[J].Appl Sci Res,1975,31:377-395.
[3]KAUP D J.A higher-order water-wave equation and the method for solving it[J].Progr Theoret Phys,1975,54:396-408.
[4]KUPERSHMIDT B A.Mathematics of dispersive water waves[J].Commun Math Phys,1985,99:51-73.
[5]YAN Zhenya,ZHANG Hongqing.New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics[J].Phys Lett A,1999,252:291-296.
[6]XIE Fuding,YAN Zhenya,ZHANG Hongqing.Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations[J].Phys Lett A,2001,285:76-80.
[7]FAN Engui,ZHANG Hongqing.A new approach to Bcklund transformations of nonlinear evolution equations[J].Appl Math Mech,1998,19:645-650.
[8]LISLE I G,REID G J.Cartan structure of infinite Lie pseudo-groups[C]∥Geometrical Approaches to Differential Equations,PJVassiliou and IG Lisle eds,2000:116-145.
[9]LISLE I G,REID G J,BOULTON A.Algorithmic determination of the structure of infinite symmetry groups of differential equations[C]∥Proceedings of the 1995International Symposium on Symbolic and Algebraic Computation,1995.
[10]REID G J,LISLE I G,BOULTON A,et al.Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs[C]∥Proceedings of the 1992International Symposium on Symbolic and Algebraic Computation,1992:63-68.
[11]REID G J.Finding abstract Lie symmetry algebras of differential equations without integrating determining equations[J].Euro JAppl Math,1991,2:319-340.
[12]OLVER P J,POHJANPELTO J.Maurer-Cartan forms and the structure of Lie pseudo-groups[J].Selecta Math,2005,11:99-126.
[13]FELS M,OLVER P J.Moving coframesⅠ.A practical algorithm[J].Acta Appl Math,1998,51:161-213.
[14]FELS M,OLVER P J.Moving coframesⅡ.Regularization and theoretical foundations[J].Acta Appl Math,1999,55:127-208.
[15]OLVER P J.Equivalence,invariants and symmetry[M].Cambridge:Cambridge University Press,1995.
[16]OLVER P J.Applications of lie groups to differential equations.second ed.graduate texts in mathe-matics[M].New York:SpringerVerlag,1993.
[17]KOGAN I A,OLVER P J.Invariants of objects and their images under surjective maps[J].Lobachevskii J Math,2015,36:260-285.
[18]OLVER P J.The symmetry groupoid and weighted signature of a geometric object[J].J Lie Theory,2015,26:235-267.
[19]OLVER P J.Recursive moving frames[J].Results Math,2011,60:423-452.
[20]OLVER P J.Differential invariant algebras[J].Contemp Math,2011,549:95-121.
[21]ROBERTT,FRANCIS V.Group foliation of finite difference equations[J].CommunNonlinear Sci,2018,59:235-254.