Journal of Qufu Normal University, Volume. 51, Issue 3, 52(2025)
One kind of integrable coupled system and its bi-Hamiltonian structures on time scale
[1] [1] FOKAS A S, GEL'FAND I M. Bi-Hamiltonian Structures and Integrability[M]. Berlin: Springer, 1993.
[2] [2] ABLOWITZ M J, SEGUR H. Solitons and the Inverse Scattering Transform[M]. New York: SIAM, 1982.
[3] [3] BASZAK M. Multi-Hamiltonian Theory of Dynamical Systems[M]. Berlin: Springer, 1998.
[4] [4] GEL'FAND I M, DICKEY L A. Fractional powers of operators and Hamiltonian systems[J]. Funct Anal Appl, 1976, 10:259-273.
[5] [5] MANIN Y I. Algebraic aspects of nonlinear differential equations[J]. Soviet Math, 1979, 11 :1-122.
[6] [6] REYMAN A G, SEMENOV M A. Family of Hamiltonian structures, hierarchy of Hamiltonians and reduction for matrix first order-differential operators[J]. Funkz Analys Priloz, 1980, 14:77-78.
[7] [7] SEMENOV M A. What is a classical R-matrix[J]? Funct Anal Appl, 1983, 17:259-272.
[8] [8] LI L C, PARMENTIER S. Nonlinear Poisson structures and R-matrices[J]. Commun Math Phys, 1989, 125:545-563.
[9] [9] ADLER M. On a trace functional for pseudo differential operators and the symplectic structure of the Kortewegde Vries equation[J]. Invent Math, 1979, 50:219-248.
[10] [10] KONOPELCHENKO B G, OEVEL W. An R-matrix approach to nonstandard classes of integrable equations [J]. Publ Rims Kyoto Univ, 1993, 29:581-666.
[11] [11] OEVEL W, RAGNISCO O. R-matrices and higher Poisson brackets for integrable systems[J]. Physica A, 1989, 161 :181-220.
[12] [12] BASZAK M, MARCINIAK K. R-matrix approach to lattice integrable systems[J]. Math Phys, 1994, 35:46-64.
[13] [13] BASZAK M. R-matrix approach to multi-Hamiltonian Lax dynamics[J]. Rep Math Phys, 1997, 40:395-430.
[14] [14] BASZAK M, SILINDIR B, SZABLIKOWSKI B M. The R-matrix approach to integrable systems on time scales[J]. Phys A Math Theor, 2008, 41 :385203.
[15] [15] HILGER S. Analysis on measure chains: A unified approach to continuous and discrete calculus[J]. Results Math, 1990, 18:18-56.
[16] [16] BOHNER M, PETERSON A. Dynamic Equations on Time Scales an Introduction with Applications [M]. Boston: Birkhauser, 2001.
[17] [17] AGARWAL R P, BOHNER M, PETERSON A. Inequalities on time scales: A survey[J]. Math Inequal Appl, 2003, 4:535-558.
[18] [18] SHENG Q, FADAG M, HENDERSON J, et al. An exploration of combined dynamic derivatives on time scales and their applications[J]. Nonlinear Anal Real, 2006, 7:395-413.
[19] [19] ATICI F M, BILES D C, LEBEDINSKY A. An application of time scales to economics[J]. Math Comput Model, 2006, 43:718-726.
[20] [20] ATICI F M, UYSAL F A. Production-inventory model of HMMS on time scales[J]. Appl Math Lett, 2008, 21 :236-243.
[21] [21] TISDELL C, ZAIDI A. Basic qualitative and quantitative results for solutionsto nonlinear, dynamic equations on time scales with an application to economic modelling I[J]. Nonlinear Anal Theor, 2008, 68（11）:3504-3524.
[22] [22] ZHANG Z, FENG R. Oscillation of third-order nonlinear emden-fowler delay dynamic equation with a sublinear neutral term on time scales[J]. Adv Differ Equ-ny, 2021, 1-14.
[23] [23] LIU Q P. Bi-Hamiltonian structures of the coupled AKNS hierarchy and the coupled Yajima-Oikawa hierarchy[J]. Math Phys, 1996, 37:2307-2314.
[24] [24] BASZAK M, SILINDIR B, SZABLIKOWSKI B M. Bi-Hamiltonian structures for integrable systems on regular time scales[J]. Math Phys, 2009, 50:073502.
[25] [25] GRSES M, GUSEINOV G SH, SILINDIR B. Integrable equations on time scales[J]. Math Phys, 2005, 46:113510.
Get Citation
Copy Citation Text
CHENG Rumeng, DONG Huanhe. One kind of integrable coupled system and its bi-Hamiltonian structures on time scale[J]. Journal of Qufu Normal University, 2025, 51(3): 52
Received: Nov. 11, 2023
Accepted: Aug. 25, 2025
Published Online: Aug. 25, 2025
The Author Email: DONG Huanhe (donghuanhe@126.com)
