New multi-symplectic Fourier pseudospectral method for a class of DGH equation

Abstract

Abstract: A numerical solution method of DGH equation was studied based on multi-symplectic theory in Hamilton space system. The multi-symplectic format of DGH equation was constructed with Fourier pseudospectral method and this format met the multi-symplectic conservation law. It was shown by the result of a numeric computation example that this multi-symplectic discrete format would have a better numeric value stability for a long time.

Key words: Hamilton system, Fourier pseudospectral method, multi-symplectic algorithm, DGH equation

CLC Number:

DAI Hong-bin, WANG Jun-jie, CAI Shan-shan, HOU Yuan-yuan. New multi-symplectic Fourier pseudospectral method for a class of DGH equation[J]. Journal of Lanzhou University of Technology, 2020, 46(1): 162-166.

References

[1] DULIN R,GOTTWALD G,HOLM D.An integrable shallow water equation with linear and nolinear dispersion [J].Physical Review Letters,2011,9:4501-4504.
[2] CAMASSA R,HOLM D.An integrable shallow water equation with peaked solutions [J].Physical Review Letters,1993,11:1661-1664.
[3] FUCHSSTEINER B,HOLM D.Symplectic structures their Backlund transformation and hereditary symmetries [J].Physica D,1981,4:47-66.
[4] TIAN L,SHI Q.Boundary control of viscous Dullin-Gottwald-Holm equation [J].International Journal of Nonlinear Science,2007,4(1):67-75.
[5] CUNG T A,PHAN T T.Decay characterization of solutions to the viscous Camassa-Holm equations [J].Nonlinearity,2018,31(2):621-650.
[6] MI Y,GUO B,MU C.Lower order regularity for the generalized Camassa-Holm equation [J].Applicable Analysis,2017,97(7):1126-1137.
[7] NOVUZOV E.Blow-up of solutions for the dissipative Dullin-Gottwald-Holm equation with arbitrary coefficients [J].Journal of Differential Equations,2016,261(2):1115-1127.
[8] DMITRY S,LECH Z.The inverse scattering transform in theform of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation [J].Opuscula Mathematica,2017,37(1):167-187.
[9] FENG K,QIN M.The symplectic methods for the computation of Hamiltonian equations [M].Berlin:Springer,1987.
[10] KONG L,LIU R,ZHENG X.A survey on symplectic and multi-symplectic algorithms [J].Applied Mathematics and Computation,2007,186:670-684.
[11] ASCHER U,MCLACHLAN R.Multisymplectic box schemes and the Korteweg-de Vries equation [J].Applied Numerical Mathematics,2004,48:255-269.
[12] 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.
[13] ESCHER J,LECHTENFELD O,YIN Z.Well-posedness and blow-up phenomena for the 2-Component Camassa-Holm equation [J].Discrete and Continous Dynamical Systems,2007,19:493-513.
[14] WANG J.Multi-symplectic Fourier pseudospectral method for a higher order wave equation of KdV type [J].Journal of Computational Mathematics,2015,33(4):379-395.
[15] ISLAS A,SCHOBER C.Backward error analysis for multisymplectic discretization of Hamiltonian PDEs [J].Mathematics and Computers in Simulation,2005,69:290-303.
[16] MOORE B,REICH S.Backward error analysis for multi-symplectic integration methods [J].Numerische Mathematik,2003,95:625-652.
[17] WANG J.Multisymplectic Fourier pseudospectral method for the nonlinear schrdinger equation with wave operator [J].Journal of Computational Mathematics,2007,25(1):31-48.
[18] JONATHAN H.Noether's theorem in multisymplectic geometry [J].Differential Geometry and its Applications,2018,56:260-294.
[19] LEON M,PRIETO M,PEDRO D,et al.Hamilton-Jacobi theory in multisymplectic classical field theories [J].Journal of Mathematical Physics,2017,58(9):1-36.
[20] COHEN D,VERDIER O.MultiSymplectic discretization of wave map equations [J].SIAM Journal on Scientific Computing,2016,38(2):A953-A972.
[21] 殷久利,田立新.一类非线性色散方程中的新型奇异孤立波 [J].物理学报,2009,58(6):3632-3636.