Whitham-Broer-Kaup方程的微分不变方程
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摘要
研究海洋科学、非线性动力学、数学物理等领域的重要模型Whitham-BroerKaup方程.Whitham-Broer-Kaup方程的微分不变方程和正规化微分不变量在运用微分不变量求解Whitham-Broer-Kaup方程过程中起到重要作用.由于该方程非线性的复杂性和经典活动标架法的受限性,运用最新的等变活动标架理论,结合Maple软件求解了难以求解出的该方程的微分不变方程和正规化微分不变量.此方法突破了以往方法的局限,仅使用无穷小决定方程组,选取适当截面,不再受限于经典活动标架法的几何范围,在计算上具有高效性和可操作性.得到的结果能够用于利用微分不变量寻找WhithamBroer-Kaup方程的解和不变性质以及研究水波、大气等非线性运动的本质.
The research object of this paper is Whitham-Broer-Kaup equation which is an important model,in oceanic science,nonlinear dynamics,mathematical physics and other fields.The differential invariant equation and normalized differential invariants of the Whitham-Broer-Kaup equation plays a very important role in solving the Whitham-Broer-Kaup equation with differential invariants.Due to the nonlinear complexity of this equation by and the limitation of classic moving frames,it's difficult to find the differential invariant equation and normalized differential invariants of this equation.The new equivariant moving frames theory and Maple software are used to solve differential invariant equation and normalized differential invariants of Whitham-Broer-Kaup equation in this paper.This algorithm breaks the limitation of previous methods.Only the infinitesimal determining equations and suitable cross-section are used.It is not limited to the geometry range of classic moving frame and is efficient and operative in computation.These conclusions are useful to find the solutions and its invariant propeties of Whitham-Broer-Kaup equation by using differential invariant and analyze the nature of nonlinear motion in oceanic science and atmospheric science.
The research object of this paper is Whitham-Broer-Kaup equation which is an important model,in oceanic science,nonlinear dynamics,mathematical physics and other fields.The differential invariant equation and normalized differential invariants of the Whitham-Broer-Kaup equation plays a very important role in solving the Whitham-Broer-Kaup equation with differential invariants.Due to the nonlinear complexity of this equation by and the limitation of classic moving frames,it's difficult to find the differential invariant equation and normalized differential invariants of this equation.The new equivariant moving frames theory and Maple software are used to solve differential invariant equation and normalized differential invariants of Whitham-Broer-Kaup equation in this paper.This algorithm breaks the limitation of previous methods.Only the infinitesimal determining equations and suitable cross-section are used.It is not limited to the geometry range of classic moving frame and is efficient and operative in computation.These conclusions are useful to find the solutions and its invariant propeties of Whitham-Broer-Kaup equation by using differential invariant and analyze the nature of nonlinear motion in oceanic science and atmospheric science.
引文
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[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[J].Geometrical Approaches to Differential Equations,2000,15:116-145.
[9]LISLE I G,REID G J,BOULTON A.Algorithmic determination of the structure of infinite symmetry groups of differential equations[C]∥In 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.California:Academic Press,1992:63-68.
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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.
[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,1995:105-208.
[16]OLVER P J.Applications of Lie groups to differential equations graduate texts in mathematics[M].New York:Springer-Verlag,1993:75-238.
[17]李玮,苏忠玉,李文爽,等.Camassa-Holm方程的活动标架应用[J].辽宁师范大学学报(自然科学版),2018,41(3):321-325.
[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[J].Geometrical Approaches to Differential Equations,2000,15:116-145.
[9]LISLE I G,REID G J,BOULTON A.Algorithmic determination of the structure of infinite symmetry groups of differential equations[C]∥In 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.California:Academic Press,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,1995:105-208.
[16]OLVER P J.Applications of Lie groups to differential equations graduate texts in mathematics[M].New York:Springer-Verlag,1993:75-238.
[17]李玮,苏忠玉,李文爽,等.Camassa-Holm方程的活动标架应用[J].辽宁师范大学学报(自然科学版),2018,41(3):321-325.