广义Broer-Kaup-Kupershmidt孤子方程的拟周期解

数学物理学报 ›› 2016, Vol. 36 ›› Issue (2): 317-327.

广义Broer-Kaup-Kupershmidt孤子方程的拟周期解

魏含玉¹, 夏铁成²

1. 周口师范学院数学与统计学院河南周口 466001;
2. 上海大学数学系上海 200444

收稿日期:2015-09-06 修回日期:2016-01-03 出版日期:2016-04-25 发布日期:2016-04-25
通讯作者: 魏含玉,weihanyu8207@163.com E-mail:weihanyu8207@163.com
基金资助:
国家自然科学基金(11271008,61072147,11547175,11447220);上海大学一流学科;河南省自然科学基金(152300410230);河南省高等学校重点科研项目(16A110026)和周口师范学院博士科研基金项目(ZKNU2014130)资助

Quasi-Periodic Solution of the Generalized Broer-Kaup-Kupershmidt Soliton Equation

Wei Hanyu¹, Xia Tiecheng²

1 College of Mathematics and Statistics, Zhoukou Normal University, Henan Zhoukou 466001;
2 Department of Mathematics, Shanghai University, Shanghai 200444

Received:2015-09-06 Revised:2016-01-03 Online:2016-04-25 Published:2016-04-25
Supported by:
Supported by the NSFC (11271008, 61072147, 11547175, 11447220), the First-class Discipline of University in Shanghai, the Science and Technology Department of Henan Province (152300410230), the Key Scientific Research Projects of Henan Province (16A110026) and the Doctoral Research Fundation of Zhoukou Normal University (ZKNU2014130)

摘要/Abstract

摘要：

该文从新谱问题出发,得到一个新的(2+1)-维广义Broer-Kaup-Kupershmidt孤子方程在Lax对非线性化下被分解成可积的常微分方程.接着,给出了一个有限维Hamilton系统并且证明在Liouville意义下是完全可积的.通过引进Abel-Jacobi坐标把Hamilton流进行了拉直,借助Riemann θ函数得到了(2+1)-维Broer-Kaup-Kupershmidt孤子方程的拟周期解.

关键词: 非线性化, Abel-Jacobi坐标, Riemann &theta, 函数, 拟周期解

Abstract:

In this paper, starting from a new spectral problem, a new (2+1)-dimensional generalized Broer-Kaup-Kupershmidt soliton equation is decomposed into systems of integrable ordinary differential equations resorting to the nonlinearization of Lax pairs. Then, a finite-dimensional Hamiltonian system is obtained and is proved to be completely integrable in Liouville sense. The Abel-Jacobi coordinates are constructed to straighten out the Hamiltonian flows, from which the quasi-periodic solution of the (2+1)-dimensional generalized Broer-Kaup-Kupershmidt soliton equation is obtained in terms of Riemann theta functions.

Key words: Nonlinearization, Abel-Jacobi coordinates, Riemann theta function, Quasi-periodic solution

中图分类号:

O175.29

魏含玉, 夏铁成. 广义Broer-Kaup-Kupershmidt孤子方程的拟周期解[J]. 数学物理学报, 2016, 36(2): 317-327.

Wei Hanyu, Xia Tiecheng. Quasi-Periodic Solution of the Generalized Broer-Kaup-Kupershmidt Soliton Equation[J]. Acta mathematica scientia,Series A, 2016, 36(2): 317-327.

参考文献

[1] Ablowitz M J, Segur H. Solitons and the Inverse Scattering Transform. Philadelphia:SIAM, 1981
[2] Novikov S, Manakov S V, Pitaevskii L P, Zakharov V E. Theory of Solitons, the Inverse Scattering Methods. New York:Consultants Bureau, 1984
[3] Cao C W, Geng X G, Wang H Y. Algebro-geometric solution of the 2+1 dimensional Burgers equation with a discrete variable. J Math Phys, 2002, 43:621-643
[4] Pan H F, Xia T C, Chen D Y. A two-component generalization of Burgers' equation with quasi-periodic solution. Rep Math Phys, 2014, 74:235-250
[5] 孙玉娟, 丁琦, 梅建琴, 张鸿庆. D-AKNS方程的代数几何解. 数学物理学报, 2013, 33A(2):276-284 Sun Y J, Ding Q, Mei J Q, Zhang H Q. Algebro-gemetric solutions of the D-AKNS equations. Acta Math Sci, 2013, 33A(2):276-284
[6] Hu X B, Wu Y T. Application of the Hirota bilinear formalism to a new integrable differential-difference equation. Phys Lett A, 1998, 246:523-529
[7] Hu X B, Ma W X. Application of Hirota's bilinear formalism to the Toeplitz lattice-some special soliton-like solutions. Phys Lett A, 2002, 293:161-165
[8] Hu X B, Tam H W. Application of Hirota's bilinear formalism to a two-dimensional lattice by Leznov. Phys Lett A, 2000, 276:65-72
[9] Cao C W. Nonlinearization of the Lax system for AKNS hierarchy. Sci China (A), 1990, 33:528-536
[10] Cao C W, Wu Y T, Geng X G. Relation between the Kadometsev-Petviashvili equation and the confocal involutive system. J Math Phys, 1999, 40:3948-3970
[11] 周汝光. 一个求孤子方程有限带势解的方法. 数学物理学报, 1998,18A(2):228-234 Zhou R G. A method to seek the finite-band solution of soliton equation. Acta Math Sci, 1998,18A(2):228-234
[12] Dai H H, Fan E G. Variable separation and algebro-geometric solutions of the Gerdjikov-Ivanov equation. Chaos Soliton Fract, 2004, 22:93-101
[13] Hon Y C, Fan E G. An algebro-geometric solution for a Hamiltonian system with application to dispersive long wave equation. J Math Phys, 2005, 46:032701
[14] Yue C, Xia T C. Algebro-geometric solutions for the complex Sharma-Tasso-Olver hierarchy. J Math Phys, 2014, 55:083511
[15] He G L, Geng X G, Wu L H. Algebro-geometric quasi-periodic solutions to the three-wave resonant interaction hierarchy. SIAM J Math Anal, 2014, 46:1348-1384
[16] Wang Z Y, Zhang J S. Explicit solutions for a new (2+1)-dimensional coupled mKdV equation. Commun Theor Phys, 2008, 49:396-400
[17] Ma W X. A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction. Chinese Ann Math Ser A, 1992, 13:115-123
[18] Yang H X, Sun Y P. Hamiltonian and super-Hamiltonian extensions related to Broer-Kaup-Kupershmidt system. Int J Theor Phys, 2010, 49:349-364
[19] Arnold V I. Mathematical Methods of Classical Mechanics. Berlin:Springer, 1978
[20] Griffiths P, Harris J. Principles of Algebraic Geometry. New York:Wiley, 1994
[21] Mumford D. Tata Lectures on Theta I, II. Boston:Birkhauser, 1984
[22] Tracy E R, Chen H H, Lee Y C. Study of quasiperiodic solutions of the nonlinear schrödinger equation and the nonlinear modulational instability. Phys Rev Lett, 1984, 53:218-221