A second order wave equation of KdV type, a important nonlinear wave equation, has broad application prospect. The equation was studied based on the multi-symplectic theory in Hamilton space. The multi-symplectic Fourier pseudospectral method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the second order wave equation of KdV type. The multi-symplectic scheme of the second order wave equation of KdV type with several conservation laws are presented. The numerical results for the second order wave equation of KdV type are reported, showing that the multi-symplectic Fourier pseudospectral method is an efficient algorithm with excellent long-time numerical behaviors.
. MULTI-SYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR A SECOND ORDER WAVE EQUATION OF KDV TYPE[J]. Journal of Numerical Methods and Computer Applicat, 2014, 35(4): 241-254.
Tzirtzilakis E, Marinakis V, Apokis C, etal. Soliton-Like solutions of higher order wave equationsof the Korteweg-de-Vries type[J]. Journal of Mathematical Physics, 2002, 43(12): 6151-6161.
Tzirtzilakis E, Xenos M, Marinakis V, etal. Interactions and stability of solitary waves in shallowwater[J], Chaos, Solitons and Fractals, 2002, 14(1): 87-95.
Fokas A S. On a class of physically important integrable equations[J]. Physics D, 1995, 87: 145-150.
Long Y, Li J, Rui W and He B. Traveling wave solutions for a second order wave equation of KdVtype[J]. Applied Mathematics and Mechanics, 2007, 28(11): 1455-1465.
Khuri S A. Soliton and periodic solutions for higher order wave equations of KdV type (I)[J].Chaos, Solitons Fractals, 2005, 26(1): 25-32.
Hong W P. Dynamics of solitary-waves in the higher order Korteweg-de Vries equation type (I)[J].Zeitschrift fur Naturforschung, 2005, 60(11): 757-767.
Li J, Rui W, Long Y and He B. Travelling wave solutions for higher-order wave equations of KdVtype(III)[J]. Mathematical Biosciences and Engineering, 2006, 3(1): 125-135.
Li J. Exact explicit peakon and periodic cusp wave solutions for several nonlinear wave equations[J]. Journal of Dynamics and Differential Equations, 2008, 20(4): 909-922.
Rui W, Long Y and He B. Some new travelling wave solutions with singular or nonsingularcharacter for the higher order wave equation of KdV type (III)[J]. Nonlinear Analysis: Theory,Methods Applications, 2009, 70(11): 3816-3828.
Marinakis V. New solutions of a higher order wave equation of the KdV type[J]. Journal ofNonlinear Mathematical Physics, 2007, 14(4): 519-525.
Rui W, Long Y and He B. Integral bifurcation method combined with computer for solving ahigher order wave equation of KdV type[J]. International Journal of Computer Mathematics Vol,2010, 87(1): 119-128
Feng K, Qin M. The symplectic methods for the computation of Hamiltonian equations[C]. Berlin:Springer, 1987.
Liu T, Qin M. Multi-symplectic geometry and multi-symplectic preissmann scheme for the KPequation[J]. Journal of Mathematical Physics, 2002, 43(8): 4060-4077.
Hu W, Deng Z, Han S, Fan W. Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation[J]. Applied Mathematics and Mechanics, 2009, 30(8): 1027-1034.
Wang Y, Wang B, Qin M. Concatenating construction of multi-symplectic scheme for 2+1 dimensional sine-Gordon equation[J]. Science in china (series A), 2004, 47(1): 18-30.