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Camassa-Holm方程的拟周期解及其渐近行为

王振 秦玉鹏 邹丽 马瑞芳 朱贵勋

王振, 秦玉鹏, 邹丽, 马瑞芳, 朱贵勋. Camassa-Holm方程的拟周期解及其渐近行为[J]. 应用数学和力学, 2015, 36(9): 990-1002. doi: 10.3879/j.issn.1000-0887.2015.09.010
引用本文: 王振, 秦玉鹏, 邹丽, 马瑞芳, 朱贵勋. Camassa-Holm方程的拟周期解及其渐近行为[J]. 应用数学和力学, 2015, 36(9): 990-1002. doi: 10.3879/j.issn.1000-0887.2015.09.010
WANG Zhen, QIN Yu-peng, ZOU Li, MA Rui-fang, ZHU Gui-xun. Quasi-Periodic Solution and Its Asymptotic Behavior for Camassa-Holm Equation[J]. Applied Mathematics and Mechanics, 2015, 36(9): 990-1002. doi: 10.3879/j.issn.1000-0887.2015.09.010
Citation: WANG Zhen, QIN Yu-peng, ZOU Li, MA Rui-fang, ZHU Gui-xun. Quasi-Periodic Solution and Its Asymptotic Behavior for Camassa-Holm Equation[J]. Applied Mathematics and Mechanics, 2015, 36(9): 990-1002. doi: 10.3879/j.issn.1000-0887.2015.09.010

Camassa-Holm方程的拟周期解及其渐近行为

doi: 10.3879/j.issn.1000-0887.2015.09.010
基金项目: 国家自然科学基金(51379033;50921001);国家重点基础研究计划(2013CB036101;2010CB32700);中央高校基本科研业务费专项资金(DUT2015LK34;DUT2015LK45)
详细信息
    作者简介:

    王振(1981—),男,山东人,副教授,博士(通讯作者. E-mail: wangzhen@dlut.edu.cn).

  • 中图分类号: O29;O368

Quasi-Periodic Solution and Its Asymptotic Behavior for Camassa-Holm Equation

Funds: The National Natural Science Foundation of China(51379033;50921001)
  • 摘要: 近20年来,浅水波模型Camassa-Holm(CH)方程受到诸多研究者关注。在之前的工作中,通过Hirota双线性方法得到了CH方程的单周期解.基于此,该文将对N=2时CH方程的拟周期解及其渐近行为进行研究.首先,回顾了坐标变换,扩展的双线性形式和Riemann(黎曼)θ-函数等内容,并在此基础上利用Hirota双线性方法构造了在N=2时CH方程的含有多个参数的拟周期解,并且此拟周期解是由Riemannθ-函数表示的。其次,发现了此拟周期解渐近行为的一个特点,即CH方程的此拟周期解可以退化为其二孤子解.
  • [1] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons[J].Physical Review Letters,1993,71(11): 1661-1664.
    [2] Camassa R, Holm D D, Hyman J M. A new integrable shallow water equation[J].Advances in Applied Machanics,1994,31: 1-33.
    [3] QIAO Zhi-jun. The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold commun[J].Communications in Mathematical Physics,2003,239(1/2): 309-341.
    [4] Fuchssteiner B, Fokas A S. Symplectic structures, their Bcklund transformations and hereditary symmetries[J].Physica D: Nonlinear Phenomena,1981,4(1): 47-66.
    [5] Johnson R S. Camassa-Holm, Korteweg-de Vries and related models for water waves[J].Journal of Fluid Mechanics,2002,455: 63-82.
    [6] Constantin A, Lannes D. The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations[J].Archive for Rational Mechanics and Analysis,2009,192(1): 165-186.
    [7] Parker A. On exact solutions of the regularized long-wave equation: a direct approach to partially integrable equations—I: solitary wave and solitons[J].Journal of Mathematical Physics,1995,36(7): 3498-3505.
    [8] Schiff J. The Camassa-Holm equation: a loop group approach[J].Physica D: Nonlinear Phenomena,1998,121(1/2): 24-43.
    [9] Constantin A. On the scattering problem for the Camassa-Holm equation[J].Proceedings of The Royal Society A,2001,457(2008): 953- 970.
    [10] Parker A. On the Camassa-Holm equation and a direct method of solution—I: bilinear form and solitary waves[J].Proceedings of The Royal Society A,2004,460(2050): 2929-2957.
    [11] Parker A. On the Camassa-Holm equation and a direct method of solution—II: soliton solutions[J].Proceedings of The Royal Society A,2005,461(2063): 3611-3632.
    [12] Parker A. On the Camassa-Holm equation and a direct method of solution—III:N -soliton solutions[J].Proceedings of The Royal Society A,2005,461(2064): 3893-3911.
    [13] Fan E G, Hon Y C. Quasiperiodic waves and asymptotic behavior for Bogoyavlenskii’s breaking soliton equation in (2+1) dimensions[J].Physical Review E,2008,78(3): 036607.
    [14] MA Wen-xiu, ZHOU Ru-guang, GAO Liang. Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2+1) dimensions[J].Modern Physics Letters A,2009,24(21): 1677-1688.
    [15] DAI Hui-hui, LI Yi-shen, SU Ting. Multi-soliton and multi-cuspon solutions of a Camassa-Holm hierarchy and their interactions[J].Journal of Physics A: Mathematical and Theoretical,2009,42(5): 055203.
    [16] DAI Hui-hui , LI Yi-shen. The interaction of the 〖WT5”BZ〗ω〖WT5”B4〗-soliton and 〖WT5”BZ〗ω〖WT5”B4〗-cuspon of the Camassa-Holm equation[J].Journal of Physics A: Mathematical and General,2005,38(42): 685-694.
    [17] Parker A. Cusped solitons of the Camassa-Holm equation—I: cuspon solitary wave and antipeakon limit[J].Chaos, Solitons & Fractals,2007,34(3): 730-739.
    [18] Parker A. Wave dynamics for peaked solitons of the Camassa-Holm equation[J].Chaos, Solitons & Fractals,2007,35(2): 220-237.
    [19] Parker A. Cusped solitons of the Camassa-Holm equation—II: binary cuspon-soliton interactions[J].Chaos, Solitons & Fractals,2009,41(3): 1531-1549.
    [20] Parker A. A factorization procedure for solving the Camassa-Holm equation[J].Inverse Problems,2006,22(2): 599-609.
    [21] Parker A, Matsuno Y. The peakon limits of soliton solutions of the Camassa-Holm equation[J].Journal of the Physical Society of Japan,2006,75(12): 124001.
    [22] Geronimo J S, Gesztesy F, Holden H. Algebro-Geometric solutions of the Baxter-Szeg difference equation[J].Communications in Mathematical Physics,2005,258(1): 149-177.
    [23] GENG Xian-guo, CAO Ce-wen. Decomposition of the (2+1)-dimensional Gardner equation and its quasi-periodic solutions[J].Nonlinearity,2001,14(6): 1433-1452.
    [24] Constantin A. Quasi-periodicity with respect to time of spatially periodic nite-gap solutions of the Camassa-Holm equation[J].Bulletin des Sciences Mathématiques,1998,122(7): 487-494.
    [25] Constantin A, McKean H P. A shallow water equation on the circle [J].Communications on Pure and Applied Mathematics,1999,52(8): 949-982.
    [26] Gesztesy F, Holden H. Algebro-geometry solutions of the Camassa-Holm hierarchy[J]. Revista Matemática Iberoamericana,2003,19(1): 73-142.
    [27] Gesztesy F, Holden H. Real-valued algebro-geometric solutions of the Camassa-Holm bierarchy[J].Philos Trans Roy Soc A,2008,366(1867): 1025-1054.
    [28] Kalla C, Klein C. New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation[J].Proceedings of The Royal Society A,2012,468(2141): 1371-1390.
    [29] Bobenko A I, Klein C, eds.Computational Approach to Riemann Surfaces [M]. Lecture Notes in Mathematics, Vol2013. Beirlin: Springer, 2011.
    [30] Trefethen L N.Spectral Methods in Matlab [M]. Software, Environments, Tools . Philadelphina, PA: SIAM, 2000.
    [31] Nakamura A. A direct method of calculating periodic wave solitons to nonlinear evolution equations—I: exact two-periodic wave solution[J].Journal of the Physical Society of Japan,1979,47(5): 1701-1705.
    [32] Frauendiener J, Klein C. Hyperelliptic theta functions and spectral methods[J].Journal of Computational and Applied Mathematics,2004,167(1): 193-218.
    [33] WANG Zhen, ZOU Li, ZONG Zhi. Periodic solutions of the Camassa-Holm equation based on the bilinear form [J].Journal of Physics A Mathematical and Theoretical,2011,44(35): 355204.
    [34] Hirota R.The Direct Method in Soliton Theory[M]. NagaiA, Nimmo J, Gilson C, eds. Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, 2004.
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出版历程
  • 收稿日期:  2015-05-13
  • 修回日期:  2015-07-08
  • 刊出日期:  2015-09-15

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