Camassa-Holm方程的活动标架应用
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摘要
研究对象是数学物理等领域的浅水波模型Camassa-Holm方程.正规化Maurer-Cartan形式的基是寻找Camassa-Holm方程解的不变性的重要工具,由于CamassaHolm方程的非线性和经典活动标架法的局限性,该方程的正规化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,本文给出了求解CamassaHolm方程正规化Maurer-Cartan形式的基的一种有效方法.该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Camassa-Holm方程解的不变性,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 research object of this paper is the shallow water wave model—Camassa-Holm equation,in mathematical physics and other fields.The basis of the invariantized Maurer-Cartan forms is an important tool for finding the solutions' invariant properties of the Camassa-Holm equation.Due to the nonlinearity of the Camassa-Holm equation and the limitations of classic moving frames,the basis of the invariantized Maurer-Cartan forms of this equation has not yet been given as far as we know.In this paper,based on the equivariant moving frames theory,with Maple software,an efficient method is provided to obtain a basis of the invariantized Maurer-Cartan forms of Camassa-Holm equation.This method overcomes the limitation 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.The conclusions in this paper can be used to find the solutions' invariant properties of Camassa-Holm equation and further study the laws and trends of motion in oceanic science,atmospheric science,nonlinear dynamics,etc.
The research object of this paper is the shallow water wave model—Camassa-Holm equation,in mathematical physics and other fields.The basis of the invariantized Maurer-Cartan forms is an important tool for finding the solutions' invariant properties of the Camassa-Holm equation.Due to the nonlinearity of the Camassa-Holm equation and the limitations of classic moving frames,the basis of the invariantized Maurer-Cartan forms of this equation has not yet been given as far as we know.In this paper,based on the equivariant moving frames theory,with Maple software,an efficient method is provided to obtain a basis of the invariantized Maurer-Cartan forms of Camassa-Holm equation.This method overcomes the limitation 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.The conclusions in this paper can be used to find the solutions' invariant properties of Camassa-Holm equation and further study the laws and trends of motion in oceanic science,atmospheric science,nonlinear dynamics,etc.
引文
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[9]LISLE I G,REID G J.Cartan structure of infinite Lie pseudo-groups[G]∥VASSILIOU P J,LISLE I G.Geometrical approaches to differential equations,2000:116-145.
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[11]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.Cambridge:Academic Press,1992:63-68.
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[13]OLVER P J,POHJANPELTO J.Maurer-Cartan forms and the structure of Lie pseudo-groups[J].Selecta Math,2005,11:99-126.
[14]FELS M,OLVER P J.Moving coframes.Ⅰ.A practical algorithm[J].Acta Appl Math,1998,51:161-213.
[15]FELS M,OLVER P J.Moving coframes.Ⅱ.Regularization and theoretical foundations[J].Acta Appl Math,1999,55:127-208.
[16]OLVER P J.Equivalence,invariants and symmetry[M].Cambridge:Cambridge University Press,1995:175-187.
[17]OLVER P J.Applications of Lie groups to differential equations[M].2nd ed.New York:Graduate Texts in Mathematics,Springer-Verlag,1993:101-130.
[18]OLVER P J,POHJANPELTO J.Moving frames for Lie pseudo-groups[J].Canadian J Math,2008:60:1336-1386.
[2]FOKAS A,OLVER P J,ROSENAU P.A plethora of integrable biHamiltonian equations[J].Progr Nonlinear Differential Equations Appl,1997,26:93-101.
[3]VELAN M S,LAKSHMANAN M.Lie symmetries and invariant solutions of the shallow-water equation[J].Int J Non-Linear Mechanics,1996,31:339-344.
[4]CONSTANTIN A,ESCHER J.Wave breaking for nonlinear nonlocal shallow water equations[J].Acta Math,1998,181:229-243.
[5]CHRISTOV O,HAKKAEV S.On the Cauchy problem for the periodic bfamily of equations and of non-uniform continuity of Degasperis-Procesi equation[J].J Math Anal Appl,2009,360:47-56.
[6]LENELLS J.Infinite propagation speed of the Camassa-Holm equation[J].J Math Anal Appl,2007,325:1468-1478.
[7]TIAN L,GUI G,GUO B.The limit behavior of the solutions to a class of nonlinear dispersive wave equations[J].Math Anal Appl,2008,341:1311-1333.
[8]LAI S Y,WU Y H.A model containing both the Camassa-Holm and Degasperis-Procesi equations[J].J Math Anal Appl,2011,374:458-469.
[9]LISLE I G,REID G J.Cartan structure of infinite Lie pseudo-groups[G]∥VASSILIOU P J,LISLE I G.Geometrical approaches to differential equations,2000:116-145.
[10]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.Cambridge:ACM Press,1995:1-6.
[11]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.Cambridge:Academic Press,1992:63-68.
[12]REID G J.Finding abstract Lie symmetry algebras of differential equations without integrating determining equations[J].Euro J Appl Math,1991,2:319-340.
[13]OLVER P J,POHJANPELTO J.Maurer-Cartan forms and the structure of Lie pseudo-groups[J].Selecta Math,2005,11:99-126.
[14]FELS M,OLVER P J.Moving coframes.Ⅰ.A practical algorithm[J].Acta Appl Math,1998,51:161-213.
[15]FELS M,OLVER P J.Moving coframes.Ⅱ.Regularization and theoretical foundations[J].Acta Appl Math,1999,55:127-208.
[16]OLVER P J.Equivalence,invariants and symmetry[M].Cambridge:Cambridge University Press,1995:175-187.
[17]OLVER P J.Applications of Lie groups to differential equations[M].2nd ed.New York:Graduate Texts in Mathematics,Springer-Verlag,1993:101-130.
[18]OLVER P J,POHJANPELTO J.Moving frames for Lie pseudo-groups[J].Canadian J Math,2008:60:1336-1386.