广义耦合方程的延拓结构

数学物理学报 ›› 2018, Vol. 38 ›› Issue (4): 658-670.

广义耦合方程的延拓结构

白喜瑞

宁波大学非线性研究中心浙江宁波 315211

收稿日期:2017-05-31 修回日期:2017-10-16 出版日期:2018-08-26 发布日期:2018-08-26
作者简介:白喜瑞,E-mail:15729216969@163.com

Prolongation Structures of Generalized Coupled Equation

Bai Xirui

Center for Nonlinear Studies, Ningbo University, Zhejiang Ningbo 315211

Received:2017-05-31 Revised:2017-10-16 Online:2018-08-26 Published:2018-08-26

摘要/Abstract

摘要： 该文利用延拓结构理论及单（半单）Lie代数的性质，研究了两组对偶系统的延拓结构.并且利用Lie代数表示理论，给出了两组对偶系统的Lax对表示.对于Camassa-Holm（CH）型的方程，通过对其超定方程的分析，仅仅选择了阶数小于等于2的函数F进行讨论，然而经过计算与分析却只存在阶数为1的情况.

关键词: 广义耦合方程, 延拓结构, 对偶系统, Lax对

Abstract: In this paper, we adopt the theory of continuation structure and properties of simple or semi-simple Lie algebra and studied the prolongation structure for two dual systems. Furthermore, the Lax pair of these equations are obtained by the method of Lie representation theory. Considering the CH type equation, based on analysing its determined equations, we select the functions F that ord(F) ≤ 2. However, there merely exists one order situation through calculation and analysis.

Key words: Generalized coupled equation, Prolongation structure, Dual system, Lax pair

中图分类号:

O175.2

白喜瑞. 广义耦合方程的延拓结构[J]. 数学物理学报, 2018, 38(4): 658-670.

Bai Xirui. Prolongation Structures of Generalized Coupled Equation[J]. Acta mathematica scientia,Series A, 2018, 38(4): 658-670.

参考文献

[1] Olver P J, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Physical Review E, 1996, 53(2):1900-1906
[2] Wahlquist H D, Estabrook F B. Prolongation structures of nonlinear evolution equations. J Math Phys, 1975, 16(1):1-7
[3] Estabrook F B, Wahlquist H D. Prolongation structures of nonlinear evolution equations Ⅱ. J Math Phys, 1976, 17(7):1293-1297
[4] Roelofs G H M, Martini R. Prolongation structure of the KdV equation in the bilinear form of Hirota. Journal of Physics A:Mathematical and General, 1990, 23(11):1877-1884
[5] Morris H C. Prolongation structures and nonlinear evolution equations in two spatial dimensions Ⅱ:A generalized nonlinear Schrödinger equation. J Math Phys, 1977, 18(2):285-288
[6] Duan X J, Deng M, Zhao W Z, et al. The prolongation structure of the inhomogeneous equation of the reaction-diffusion type. Journal of Physics A:Mathematical and Theoretical, 2007, 40(14):3831-3837
[7] Yang Y Q, Chen Y. Prolongation structure of the equation studied by Qiao. Communications in Theoretical Physics, 2011, 56(3):463-466
[8] Wang D S, Ma Y Q, Li X G. Prolongation structures and matter-wave solitons in F=1 spinor BoseEinstein condensate with time-dependent atomic scattering lengths in an expulsive harmonic potential. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(10):3556-3569
[9] Bracken P. Exterior differential systems prolongations and application to a study of two nonlinear partial differential equations. Acta Appl Math, 2011, 113(3):247-263
[10] Stalin S, Senthilvelan M. A note on the prolongation structure of the cubically nonlinear integrable Camassa-Holm type equation. Physics Letters A, 2011, 375(43):3786-3788
[11] 白永强, 裴明, 高文娟. Camassa-Holm和Degasperis-Procesi方程的延拓结构(英文). 河南大学学报(自然科学版), 2011, 41(6):551-555 Bai Y Q, Pei M, Gao W J. Prolongation structures of the Camassa-Holm and Degasperis-Procesi equations. Journal of Henan University (Natural Science), 2011, 41(6):551-555
[12] Bracken P. An exterior differential system for a generalized Korteweg-de Vries equation and its associated integrability. Acta Appl Math, 2007, 95(3):223-231
[13] Wang D S. Integrability of a coupled KdV system:Painlevé property, Lax pair and Bäcklund transformation. Applied Mathematics and Computation, 2010, 216(4):1349-1354
[14] Wang D S. Complete integrability and the Miura transformation of a coupled KdV equation. Applied Mathematics Letters, 2010, 23(6):665-669
[15] Cao Y H, Wang D S. Prolongation structures of a generalized coupled Korteweg-de Vries equation and Miura transformation. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(9):2344-2349
[16] Wang D S. Integrability of the coupled KdV equations derived from two-layer fluids:Prolongation structures and Miura transformations. Nonlinear Analysis:Theory, Methods and Applications, 2010, 73(1):270-281
[17] Wang D S, Wei X Q. Integrability and exact solutions of a two-component Korteweg-de Vries system. Applied Mathematics Letters, 2016, 51:60-67
[18] Wang D S, Liu J, Zhang Z F. Integrability and equivalence relationships of six integrable coupled Kortewegde Vries equations. Mathematical Methods in the Applied Sciences, 2016, 39(12):3516-3530
[19] Humphreys J E. Introduction to Lie Algebras and Representation Theory. New York:Springer-Verlag, 1972
[20] Jacobson N. Lie Algebras. New York:Dover Publications, 1979
[21] Foursov M V. On integrable coupled KdV-type systems. Inverse Problems, 2000, 16(1):259-274
[22] Flanders H. Differential Forms. New York:Academic Press, 1963:1-73
[23] Dodd R, Fordy A P. The prolongation structures of quasi-polynomial flows. Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences, The Royal Society, 1983, 385(1789):389-429
[24] 罗婷. 多分量KdV, mKdV方程组的对偶系统[D]. 西安:西北大学,2013 Luo T. Dual System of Multi-Component KdV and mKdV Equations[D]. Xi'an:Northwest Univesity, 2013
[25] Fuchssteiner B. The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems. Progress of Theoretical Physics, 1982, 68(4):1082-1104