非线性Klein-Gordon-Zakharov方程组的多辛算法
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摘要
一切耗散效应可以忽略不计的物理过程都可表示成能够保持辛几何结构不变的哈密尔顿系统的形式,它在自然界中具有普适性,也就是说大多数孤子方程都可以表示成哈密尔顿形式.现代数值计算的基本原则是尽可能保持原问题的本质特征.因此,研究保持哈密尔顿系统的辛几何结构特征的数值方法是必然的.本论文主要讨论无穷维Hamilton系统的多辛几何算法.与辛几何算法的主要差别是,多辛算法不仅在一定的边界条件下保持系统的离散空间上的辛形式之和,而且能够保持局部的辛形式,从而多辛几何算法更多的是体现在系统局部的守恒性质,更能体现系统的本质特征.
本文研究的无穷维Hamilton系统中Klein-Gordon-Zakharov (简称KGZ)方程组是一个重要的模型,它是由一个Klein-Gordon方程和一个Zakharov方程耦合而成.我们通过对KGZ方程组作正则变换后,得到了它的一个多辛方程组及其几个相关守恒律.然后用Gauss-Legendre Runge-Kutta方法对此多辛方程组离散,得到了KGZ方程组的多辛格式,证明了该格式具有离散形式的多辛守恒律.对中点格式,通过消去中间变量得到了与多辛格式等价的多辛Preissman格式.我们通过大量数值实验验证了所构造的多辛格式的有效性和长时间的数值稳定性,同时多辛格式还能很好地模拟原孤立波的波形,说明我们的理论分析是正确的.
另外,本文还通过对空间和时间方向分别用Fourier拟谱方法和中点方法离散KGZ方程组的多辛方程组,得到了非线性KGZ方程组的多辛Fourier拟谱格式.我们通过数值实验证明了该格式的有效性.数值结果表明多辛Fourier拟谱格式是正确可行的.
All the physical courses whose dissipative effects are negligible can be expressed as Hamiltonian systems which preserve energy conservation and symplectic geometric structure. The Hamiltonian system is universal in the nature, in other words, most soliton equations can be written into Hamiltonian formalism. The basic principle of modern numerical computation is to preserve the intrincal character of the original problems. Therefore, it is necessary to study numerical methods which preserve the symplectic structure of the Hamiltonian system. This thesis illustrates the multi-symplectic algorithms for infinite dimensional Hamiltonian systems. The multi-symplectic algorithms conserve not only the sum of symplectic forms on discrete space under appropriate boundary conditions, but also symplectic forms in local representations. Therefore, multi-symplectic algorithms have remarkable local properties, and describe the system more essentially.
In the dissertation, Klein-Gordon-Zakharov(KGZ) equations are important model of infinite dimensional Hamiltonian systems in physics, which are coupled by the Klein-Gordon equation and Zakharov equation. Multi-symplectic equations for KGZ equations are presented which posses some conservation laws by canonical transformations. Using the Gauss-Legendre Runge-Kutta method to discrete multi-symplectic equations, we present the multi-symplectic schemes for the KGZ equations. They preserve discretic multi-symplectic conservation law exactly. We eliminate middle variables and amount to a single variable multi-symplectic schemes for mid-point schemes, and call them Preissman schemes. We perform a lot of experiments to show that the schemes preserve the multi-symplectic geometry structure exactly by satisfying the discrete multi-symplectic conservation law, and can simulate the original waves in a long time. Numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results.
In other words, in the dissertation we present multi-symplectic Fourier pseudo-spectral schemes for the KGZ equations, using Fourier pseudo-spectral method and mid-point method to discrete multi-symplectic equations in space and time directions, respectively. We also carry out some experiments to illustrate their validity. Numerical results show that multi-symplectic Fourier pseudo-spectral schemes are right.
本文研究的无穷维Hamilton系统中Klein-Gordon-Zakharov (简称KGZ)方程组是一个重要的模型,它是由一个Klein-Gordon方程和一个Zakharov方程耦合而成.我们通过对KGZ方程组作正则变换后,得到了它的一个多辛方程组及其几个相关守恒律.然后用Gauss-Legendre Runge-Kutta方法对此多辛方程组离散,得到了KGZ方程组的多辛格式,证明了该格式具有离散形式的多辛守恒律.对中点格式,通过消去中间变量得到了与多辛格式等价的多辛Preissman格式.我们通过大量数值实验验证了所构造的多辛格式的有效性和长时间的数值稳定性,同时多辛格式还能很好地模拟原孤立波的波形,说明我们的理论分析是正确的.
另外,本文还通过对空间和时间方向分别用Fourier拟谱方法和中点方法离散KGZ方程组的多辛方程组,得到了非线性KGZ方程组的多辛Fourier拟谱格式.我们通过数值实验证明了该格式的有效性.数值结果表明多辛Fourier拟谱格式是正确可行的.
All the physical courses whose dissipative effects are negligible can be expressed as Hamiltonian systems which preserve energy conservation and symplectic geometric structure. The Hamiltonian system is universal in the nature, in other words, most soliton equations can be written into Hamiltonian formalism. The basic principle of modern numerical computation is to preserve the intrincal character of the original problems. Therefore, it is necessary to study numerical methods which preserve the symplectic structure of the Hamiltonian system. This thesis illustrates the multi-symplectic algorithms for infinite dimensional Hamiltonian systems. The multi-symplectic algorithms conserve not only the sum of symplectic forms on discrete space under appropriate boundary conditions, but also symplectic forms in local representations. Therefore, multi-symplectic algorithms have remarkable local properties, and describe the system more essentially.
In the dissertation, Klein-Gordon-Zakharov(KGZ) equations are important model of infinite dimensional Hamiltonian systems in physics, which are coupled by the Klein-Gordon equation and Zakharov equation. Multi-symplectic equations for KGZ equations are presented which posses some conservation laws by canonical transformations. Using the Gauss-Legendre Runge-Kutta method to discrete multi-symplectic equations, we present the multi-symplectic schemes for the KGZ equations. They preserve discretic multi-symplectic conservation law exactly. We eliminate middle variables and amount to a single variable multi-symplectic schemes for mid-point schemes, and call them Preissman schemes. We perform a lot of experiments to show that the schemes preserve the multi-symplectic geometry structure exactly by satisfying the discrete multi-symplectic conservation law, and can simulate the original waves in a long time. Numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results.
In other words, in the dissertation we present multi-symplectic Fourier pseudo-spectral schemes for the KGZ equations, using Fourier pseudo-spectral method and mid-point method to discrete multi-symplectic equations in space and time directions, respectively. We also carry out some experiments to illustrate their validity. Numerical results show that multi-symplectic Fourier pseudo-spectral schemes are right.
引文
[1] D.P.G. Drazin, R.S. Johnson, Solitons: an introduction, Cambridge University Press,1989.
[2] M.J. Ablowitz, J.F. Ladik, A nonlinear difference scheme and inverse scattering, Studies.in Appl. Math., 1981(39), 94-102.
[3] N.J.Zabusky, M.D.Kruskal,Interaction of‘soliton' in a collisionless plasma and recurrence of initial states, Phys. Rev. Lett., 1965(15), 240-243.
[4] S. Jimenez, L. Vazquez, Analysis of four numerical schemes for a nonlinear Klein-Gordon equation, Appl. Math. Comput., 1990(35), 61-94.
[5] B.Y. Guo, V.S. Manoranjan, a spectral method for solving the RLW equation, IMA JNA,1985(5), 307-318.
[6]冯康,冯康文集(II),国防工业出版社,北京,1995.
[7]冯康,秦孟兆,哈密顿系统的辛几何算法[M],浙江科学技术出版社,杭州,2002.
[8] Feng K. On difference schemes and symplectic geometry, Proceeding of the 1984 beijing symposium on D.D., Science Press, Beijing, 1985, 42-58.
[9] Feng K. Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comput. Math., 1986, 4: 279-289.
[10] Feng K, Wu H M, Qin M Z, et al. Construct of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comput. Math., 1989, 7: 71-96.
[11] Li C, Qin M Z. A symplectic difference scheme for the infinite -dimensional Hamiltonian system, J. Comput. Math., 1988, 6: 164-174.
[12] Sun G. symplectic partitioned Runge-Kutta methods, J. Comput. Math.,1993, 11: 365-372.
[13] Sun G. Construnction of high order symplectic Runge-Kutta methods, J. Comput. Math., 1993, 11: 250-260.
[14] Sun G. Construnction of high order symplectic partitioned Runge-Kutta methods, J. Comput. Math., 1995, 13: 40-50.
[15] Shang Z J. KAM theorem of symplectic algorithms for Hamiltonian systems, Numer.math., 1999, 83: 477-496.
[16] Tang Y F, Pérez-Garcia V M, Vázquez L. Symplectic methods for the Ablowitz-ladik model,Appl. Math. Comput.,1997,82:17-38.
[17] Tang Y F. The symplectic of multi-step methods, Comput. Math. Appl., 1993,25: 83-90.
[18] Tang Y F. The necessary condtion for Runge-Kutta scheme to be symplectic for Hamiltonian systems, Comput. Math. Appl.,1993, 26: 13-20.
[19] Duncan D B. Symplectic finite difference approximation of thenonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 1996, 34: 1742-1760.
[20] Yoshida H. Recent progress in the theory and application of symplectic integrators,Celestial mech.Synam. Astronom, 1993, 56: 27-43.
[21] Hairer E,Lubich C,Wanner G.Geometric numerical integration, Springer,Berlin,2002.
[22]秦孟兆.辛几何和计算哈密顿力学,力学和实践,1990,6:1-20.
[23] Qin M Z, Multi-stage symplectic schemes of two kinds of Hamiltonian systems of wave equations, Comput. math. Applic.,1990, 19: 51-62.
[24] Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems,J.Comput. math.,1991, 9: 211-221.
[25] Qin M Z, Zhu W J. Construction of higher symplectic scheme by composition,Computing,1991,47:309-321.
[26] Yoshida H. Construction of higher symplectic integrators. Phys. Lett.A,1990,150:262-268.
[27] Liu X S, Qi Y Y,He J F,et al. Recent progress in symplectic algorithms for use in quantum systems,Commun.Comput.Phys.,2007, 2: 1-53.
[28] Channell P J, Scovel C. Symplectic integration of Hamiltonian system, Nonlin-earity,1990, 3: 231-259.
[29] Lasagni F M. Canonical Runge-Kutta methods, Z. Angew. Math. Phys., 1988,39: 952-953.
[30] Sanz-Sera J M. Runge-Kutt schemes for Hamiltonian systems, BIT, 1988, 28: 877-883.
[31] Sun J Q, Gu X Y, Ma Z Q. Numerical study of the soliton waves of the coupled nonlinear Schr?dinger system. Physica D, 2004, 311-328.
[32] Liu Z, Bai Y, Li Q, et al. Symplctic and multisymplectic schemes with the simple finite element method, Phys. Lett. A, 2003, 314: 443-455.
[33] Huang M Y, Ru Q, Gong C.A structure preserving discretization of nonlinear Schr?dinger equation. J.Comput.Math., 1999, 17: 553-560.
[34] Tang Y F, Pérez-Garcia V M, Vázquez L. Symplectic methods for the Ablowitz-ladik model,Appl. Math. Comput. 1997,82:17-38.
[35] Duncan D B. Symplectic finite difference approximation of the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 1996, 34: 1742-1760.
[36] Marsden J,Patrick G P,Shkoller S.Multi-symplectic geometry, varational integrators, and nonlinear PDEs, Commun.math. Phys., 1998, 199: 351-395.
[37] Bridges T J. Multi-symplectic structures and wave propagation. Math. Proc.Camb. Phil. Soc.,1997, 121:1 47-190.
[38] Bridges T J. A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities. Proc. R.Soc. Lond. A,1997, 453: 1365-1395.
[39] Bridges T J, Reich S. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A, 2001, 284: 184-193.
[40] Reich S. Multi-symplectic Runge-Kutta method for Hamiltonian wave equations, J. Comput. Phys.,2001, 157: 473-499.
[41] Reich S. Finite volume method for Multi-symplectic PDEs, BIT. 2000, 40:559-582.
[42] Bridges T J, Reich S. Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations, Physica D, 2001, 152-153: 491-504.
[43] Bridges T J, Reich S. numerical method for Hamiltonian PDEs, J. Phys. A: Math. Gen., 2006, 39: 5287-5320.
[44] Huang L Y, Zeng W P, Qin M Z. A new multisymplectic scheme for nonlinear“good”Boussinesq equation, J. Comput. Math., 2003, 21: 703-714.
[45] Hong J L, Li C. Multisymplectic Rung-Kutta methods for nonlinear Dirac equations, J. Comput. Phys., 2006, 211: 448-472.
[46] Sun J Q, Qin M Z. Multi-symplectic methods for the coupled 1D nonlinear Schr?dinger system. Computer Physics Communications, 2003, 155: 221-235.
[47] Sun J Q, Gu X Y, Ma Z Q. Multisymplectic difference schemes for coupled nonlinear Schr?dinger system. Chinese Journal of Computional Physics, 2004, 21:321-328.
[48]孔令华.一些非线性发展方程(组)的辛和多辛算法,博士学位论文,中国科技大学,2007.
[49]江珊珊.显式与高阶多辛几何数值格式研究,博士学位论文,中国科学院研究生院,2007.
[50] Dendy R O. Plasma Dynamics. Oxford:Oxford University Press, 1990.
[51] Guo B, Yuan G. Global smooth solution for the Klein-Gordon-Zakharov equations,J. Math. Phys., 1995, 36(8): 4119-4124.
[52] Adomian G. Non-perturbative solution of the Klein-Gordon-Zakharov equations,Appl. Math. and Comput., 1997, 81(1): 89-92.
[53] Tsutaya Kimitoshi. Global existence of small amplitude solutions for the Klein-Gordon-Zakharov equations. Non. Anal., 1996, 27(12): 1373-1380.
[54] Ozawa T, Tsutaya K, Tsutsumi Y. Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with difference propagation speeds in three space dimensions. Math. Ann., 1999, 313: 127-140.
[55]陈翰林,鲜大权. Klein-Gordon-Zakharov方程组得周期波解.应用数学学报[J],2006,29(6):1139-1144.
[56] Tingchun Wang, Juan Chen, Luming Zhang, Conservative difference methods for the Klein-Gordon-Zakharov equations, Journal of Computational and Applied Mathematics, 2007,205:430-452.
[57]陈娟. Klein-Gordon-Zakharov方程的几个守恒差分格式,硕士学位论文,南京航空航天大学,2006.
[58]王廷春.某些非线性孤立波方程(组)的数值算法研究,博士学位论文,南京航空航天大学博士学位论文,2007.
[59] Miller J J H. On the location of zeros of certain classes of polynomials with application to numerical analysis. J. Inst. Math. Appls.1971,8:394-406.
[60] Richtmyer R D,Morton K W. Difference method for initial-value problems, 2nd ed. New York; Wiley.1967,59-91.
[61]李荣华,冯果忱编.微分方程数值解法[M].北京:高等教育出版社,1996.
[62]李立康等编.微分方程数值解法[M].上海:复旦大学出版社,1999.
[63]矢岛信男,野术达夫.发展方程の数值分析[M].东京:岩波书店,1977.46- 232.
[64] Liu S, Fu Z, Liu S, et al. The periodic solutions for a class of coupled nonlinear Klein-Gordon equations.Phys.Lett.A,2004,323: 415-420.
[65] Liu S, Fu Z, Liu S, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys.Lett.A,2001,289: 69-74.
[66] Jing-bo Chen,Mengzhao Qin.Multi-symplectic Fourier pseudospetral method for the nonlinear Schr?dinger Equation.ET on numerical analysis,2001.
[2] M.J. Ablowitz, J.F. Ladik, A nonlinear difference scheme and inverse scattering, Studies.in Appl. Math., 1981(39), 94-102.
[3] N.J.Zabusky, M.D.Kruskal,Interaction of‘soliton' in a collisionless plasma and recurrence of initial states, Phys. Rev. Lett., 1965(15), 240-243.
[4] S. Jimenez, L. Vazquez, Analysis of four numerical schemes for a nonlinear Klein-Gordon equation, Appl. Math. Comput., 1990(35), 61-94.
[5] B.Y. Guo, V.S. Manoranjan, a spectral method for solving the RLW equation, IMA JNA,1985(5), 307-318.
[6]冯康,冯康文集(II),国防工业出版社,北京,1995.
[7]冯康,秦孟兆,哈密顿系统的辛几何算法[M],浙江科学技术出版社,杭州,2002.
[8] Feng K. On difference schemes and symplectic geometry, Proceeding of the 1984 beijing symposium on D.D., Science Press, Beijing, 1985, 42-58.
[9] Feng K. Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comput. Math., 1986, 4: 279-289.
[10] Feng K, Wu H M, Qin M Z, et al. Construct of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comput. Math., 1989, 7: 71-96.
[11] Li C, Qin M Z. A symplectic difference scheme for the infinite -dimensional Hamiltonian system, J. Comput. Math., 1988, 6: 164-174.
[12] Sun G. symplectic partitioned Runge-Kutta methods, J. Comput. Math.,1993, 11: 365-372.
[13] Sun G. Construnction of high order symplectic Runge-Kutta methods, J. Comput. Math., 1993, 11: 250-260.
[14] Sun G. Construnction of high order symplectic partitioned Runge-Kutta methods, J. Comput. Math., 1995, 13: 40-50.
[15] Shang Z J. KAM theorem of symplectic algorithms for Hamiltonian systems, Numer.math., 1999, 83: 477-496.
[16] Tang Y F, Pérez-Garcia V M, Vázquez L. Symplectic methods for the Ablowitz-ladik model,Appl. Math. Comput.,1997,82:17-38.
[17] Tang Y F. The symplectic of multi-step methods, Comput. Math. Appl., 1993,25: 83-90.
[18] Tang Y F. The necessary condtion for Runge-Kutta scheme to be symplectic for Hamiltonian systems, Comput. Math. Appl.,1993, 26: 13-20.
[19] Duncan D B. Symplectic finite difference approximation of thenonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 1996, 34: 1742-1760.
[20] Yoshida H. Recent progress in the theory and application of symplectic integrators,Celestial mech.Synam. Astronom, 1993, 56: 27-43.
[21] Hairer E,Lubich C,Wanner G.Geometric numerical integration, Springer,Berlin,2002.
[22]秦孟兆.辛几何和计算哈密顿力学,力学和实践,1990,6:1-20.
[23] Qin M Z, Multi-stage symplectic schemes of two kinds of Hamiltonian systems of wave equations, Comput. math. Applic.,1990, 19: 51-62.
[24] Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems,J.Comput. math.,1991, 9: 211-221.
[25] Qin M Z, Zhu W J. Construction of higher symplectic scheme by composition,Computing,1991,47:309-321.
[26] Yoshida H. Construction of higher symplectic integrators. Phys. Lett.A,1990,150:262-268.
[27] Liu X S, Qi Y Y,He J F,et al. Recent progress in symplectic algorithms for use in quantum systems,Commun.Comput.Phys.,2007, 2: 1-53.
[28] Channell P J, Scovel C. Symplectic integration of Hamiltonian system, Nonlin-earity,1990, 3: 231-259.
[29] Lasagni F M. Canonical Runge-Kutta methods, Z. Angew. Math. Phys., 1988,39: 952-953.
[30] Sanz-Sera J M. Runge-Kutt schemes for Hamiltonian systems, BIT, 1988, 28: 877-883.
[31] Sun J Q, Gu X Y, Ma Z Q. Numerical study of the soliton waves of the coupled nonlinear Schr?dinger system. Physica D, 2004, 311-328.
[32] Liu Z, Bai Y, Li Q, et al. Symplctic and multisymplectic schemes with the simple finite element method, Phys. Lett. A, 2003, 314: 443-455.
[33] Huang M Y, Ru Q, Gong C.A structure preserving discretization of nonlinear Schr?dinger equation. J.Comput.Math., 1999, 17: 553-560.
[34] Tang Y F, Pérez-Garcia V M, Vázquez L. Symplectic methods for the Ablowitz-ladik model,Appl. Math. Comput. 1997,82:17-38.
[35] Duncan D B. Symplectic finite difference approximation of the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal., 1996, 34: 1742-1760.
[36] Marsden J,Patrick G P,Shkoller S.Multi-symplectic geometry, varational integrators, and nonlinear PDEs, Commun.math. Phys., 1998, 199: 351-395.
[37] Bridges T J. Multi-symplectic structures and wave propagation. Math. Proc.Camb. Phil. Soc.,1997, 121:1 47-190.
[38] Bridges T J. A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities. Proc. R.Soc. Lond. A,1997, 453: 1365-1395.
[39] Bridges T J, Reich S. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A, 2001, 284: 184-193.
[40] Reich S. Multi-symplectic Runge-Kutta method for Hamiltonian wave equations, J. Comput. Phys.,2001, 157: 473-499.
[41] Reich S. Finite volume method for Multi-symplectic PDEs, BIT. 2000, 40:559-582.
[42] Bridges T J, Reich S. Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations, Physica D, 2001, 152-153: 491-504.
[43] Bridges T J, Reich S. numerical method for Hamiltonian PDEs, J. Phys. A: Math. Gen., 2006, 39: 5287-5320.
[44] Huang L Y, Zeng W P, Qin M Z. A new multisymplectic scheme for nonlinear“good”Boussinesq equation, J. Comput. Math., 2003, 21: 703-714.
[45] Hong J L, Li C. Multisymplectic Rung-Kutta methods for nonlinear Dirac equations, J. Comput. Phys., 2006, 211: 448-472.
[46] Sun J Q, Qin M Z. Multi-symplectic methods for the coupled 1D nonlinear Schr?dinger system. Computer Physics Communications, 2003, 155: 221-235.
[47] Sun J Q, Gu X Y, Ma Z Q. Multisymplectic difference schemes for coupled nonlinear Schr?dinger system. Chinese Journal of Computional Physics, 2004, 21:321-328.
[48]孔令华.一些非线性发展方程(组)的辛和多辛算法,博士学位论文,中国科技大学,2007.
[49]江珊珊.显式与高阶多辛几何数值格式研究,博士学位论文,中国科学院研究生院,2007.
[50] Dendy R O. Plasma Dynamics. Oxford:Oxford University Press, 1990.
[51] Guo B, Yuan G. Global smooth solution for the Klein-Gordon-Zakharov equations,J. Math. Phys., 1995, 36(8): 4119-4124.
[52] Adomian G. Non-perturbative solution of the Klein-Gordon-Zakharov equations,Appl. Math. and Comput., 1997, 81(1): 89-92.
[53] Tsutaya Kimitoshi. Global existence of small amplitude solutions for the Klein-Gordon-Zakharov equations. Non. Anal., 1996, 27(12): 1373-1380.
[54] Ozawa T, Tsutaya K, Tsutsumi Y. Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with difference propagation speeds in three space dimensions. Math. Ann., 1999, 313: 127-140.
[55]陈翰林,鲜大权. Klein-Gordon-Zakharov方程组得周期波解.应用数学学报[J],2006,29(6):1139-1144.
[56] Tingchun Wang, Juan Chen, Luming Zhang, Conservative difference methods for the Klein-Gordon-Zakharov equations, Journal of Computational and Applied Mathematics, 2007,205:430-452.
[57]陈娟. Klein-Gordon-Zakharov方程的几个守恒差分格式,硕士学位论文,南京航空航天大学,2006.
[58]王廷春.某些非线性孤立波方程(组)的数值算法研究,博士学位论文,南京航空航天大学博士学位论文,2007.
[59] Miller J J H. On the location of zeros of certain classes of polynomials with application to numerical analysis. J. Inst. Math. Appls.1971,8:394-406.
[60] Richtmyer R D,Morton K W. Difference method for initial-value problems, 2nd ed. New York; Wiley.1967,59-91.
[61]李荣华,冯果忱编.微分方程数值解法[M].北京:高等教育出版社,1996.
[62]李立康等编.微分方程数值解法[M].上海:复旦大学出版社,1999.
[63]矢岛信男,野术达夫.发展方程の数值分析[M].东京:岩波书店,1977.46- 232.
[64] Liu S, Fu Z, Liu S, et al. The periodic solutions for a class of coupled nonlinear Klein-Gordon equations.Phys.Lett.A,2004,323: 415-420.
[65] Liu S, Fu Z, Liu S, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys.Lett.A,2001,289: 69-74.
[66] Jing-bo Chen,Mengzhao Qin.Multi-symplectic Fourier pseudospetral method for the nonlinear Schr?dinger Equation.ET on numerical analysis,2001.