一些非线性波动方程的多辛算法研究
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摘要
许多重要的数学物理方程都可以表示为多辛Hamilton系统的形式,从而,对其数值算法的研究无疑具有非常重要的意义。多辛几何结构是多辛Hamilton系统的内在几何性质,这就要求此结构在数值离散时得到保持,我们称能够保持此离散多辛几何结构的算法为多辛算法。大量的数值结果表明多辛算法在长时间的数值模拟中相比于非多辛算法具有明显的优越性。
本文对一些重要的一维和二维非线性波动方程的多辛算法进行了研究。主要研究成果如下:
1、分别构造了Camassa-Holm方程的多辛Fourier拟谱格式和KdV方程的多辛Fourier谱离散格式与多辛Fourier拟谱格式。此外,我们首次将Ito型耦合KdV方程化为多辛Hamilton偏微分方程的形式并构造了其多辛Fourier拟谱格式。
2、分别构造了耦合非线性Schr(o|¨)dinger方程和二维非线性Schr(o|¨)dinger方程的多辛分裂格式。
3、构造了二维多辛Hamilton偏微分方程的多辛Fourier拟谱格式并证明了该格式保持相应的离散多辛守恒律,同时将此算法应用于求解二维Zakharov-Kuznetsov方程和二维Kadomtsev- Petviashvili方程,分别构造了这两种方程的多辛Fourier拟谱格式。
4、对本文所构造的算法,我们通过大量的数值算例验证了其有效性和长时间数值模拟的优越性。
Manyimportantmathematicalphysicsequationscanbewrittenasamulti-symplecticHamiltonian system. Therefore, the research of its numerical method is of great impor-tance. Multi-symplectic structure is the intrinsic geometric property of the multisymplec-tic Hamiltonian system. It is natural to require a discretization to preserve this geometricstructure. Such a numerical algorithm is called multi-symplectic method. A great manynumerical experiments show that multi-symplectic method has significant superiority inlong time simulation compared with other method which is not multi-symplectic.
The thesis is devoted to investigate multi-symplectic methods for some one-dimens-ional and two-dimensional nonlinear wave equations. The main contributions are as fol-lows:
1. Multi-symplectic Fourier pseudospectral (MSFP) scheme are constructed for theCamassa-Holm equation. Moreover, multi-symplectic Fourier spectral discretization sch-eme and MSFP scheme for the KdV equation are constructed respectively. In addition, wefoundthattheIto-typecoupledKdVequationcanbewrittenasamulti-sympleticHamiltonpartial differential equation (PDE) and propose a corresponding MSFP scheme in the firsttime.
2. Multi-symplectic splitting methods for the coupled nonlinear Schr(o|¨)dinger equa-tion and the two-dimensional nonlinear Schr(o|¨)dinger equation are proposed respectively.
3. The MSFP scheme for two-dimensional multi-symplectic Hamiltonian PDEs isconstructed. The relevant discrete multi-symplectic conservation laws are also proved.Meanwhile, this proposed method is applied to solve the two-dimensional Zakharov-Kuznetsov equation and Kadomtsev- Petviashvili equation and the corresponding MSFPschemes are constructed.
4. A great many numerical experiments are presented to show the effectiveness andsuperiority of the proposed methods in long time simulation.
本文对一些重要的一维和二维非线性波动方程的多辛算法进行了研究。主要研究成果如下:
1、分别构造了Camassa-Holm方程的多辛Fourier拟谱格式和KdV方程的多辛Fourier谱离散格式与多辛Fourier拟谱格式。此外,我们首次将Ito型耦合KdV方程化为多辛Hamilton偏微分方程的形式并构造了其多辛Fourier拟谱格式。
2、分别构造了耦合非线性Schr(o|¨)dinger方程和二维非线性Schr(o|¨)dinger方程的多辛分裂格式。
3、构造了二维多辛Hamilton偏微分方程的多辛Fourier拟谱格式并证明了该格式保持相应的离散多辛守恒律,同时将此算法应用于求解二维Zakharov-Kuznetsov方程和二维Kadomtsev- Petviashvili方程,分别构造了这两种方程的多辛Fourier拟谱格式。
4、对本文所构造的算法,我们通过大量的数值算例验证了其有效性和长时间数值模拟的优越性。
Manyimportantmathematicalphysicsequationscanbewrittenasamulti-symplecticHamiltonian system. Therefore, the research of its numerical method is of great impor-tance. Multi-symplectic structure is the intrinsic geometric property of the multisymplec-tic Hamiltonian system. It is natural to require a discretization to preserve this geometricstructure. Such a numerical algorithm is called multi-symplectic method. A great manynumerical experiments show that multi-symplectic method has significant superiority inlong time simulation compared with other method which is not multi-symplectic.
The thesis is devoted to investigate multi-symplectic methods for some one-dimens-ional and two-dimensional nonlinear wave equations. The main contributions are as fol-lows:
1. Multi-symplectic Fourier pseudospectral (MSFP) scheme are constructed for theCamassa-Holm equation. Moreover, multi-symplectic Fourier spectral discretization sch-eme and MSFP scheme for the KdV equation are constructed respectively. In addition, wefoundthattheIto-typecoupledKdVequationcanbewrittenasamulti-sympleticHamiltonpartial differential equation (PDE) and propose a corresponding MSFP scheme in the firsttime.
2. Multi-symplectic splitting methods for the coupled nonlinear Schr(o|¨)dinger equa-tion and the two-dimensional nonlinear Schr(o|¨)dinger equation are proposed respectively.
3. The MSFP scheme for two-dimensional multi-symplectic Hamiltonian PDEs isconstructed. The relevant discrete multi-symplectic conservation laws are also proved.Meanwhile, this proposed method is applied to solve the two-dimensional Zakharov-Kuznetsov equation and Kadomtsev- Petviashvili equation and the corresponding MSFPschemes are constructed.
4. A great many numerical experiments are presented to show the effectiveness andsuperiority of the proposed methods in long time simulation.
引文
[1] Hairer E, Lubich C, Wanner G. Geometric numerical integration[M]. New York:Springer-Verlag, 2002.
[2]冯康,秦孟兆.哈密尔顿系统的辛几何算法[M].杭州:浙江科学技术出版社,2002.
[3] FengK,QinM. SymplecticgeometricalgorithmsforHamiltoniansystems[M]. NewYork: Springer-Verlag, 2009.
[4] Bridges T J. Multi-symplectic structures and wave propagation[J]. Math. Proc.Camb. Phil. Soc., 1997, 121:147-190.
[5] Bridges T J, Reich S. Multi-symplectic integrators: numerical schemes for Hamil-tonian PDEs that conserve symplecticity[J]. Phys. Lett. A, 2001, 284:184-193.
[6] Marsden J E, Patrick G W, Shkoller S. Multisymplectic geometry, variational inte-grators, and nonlinear PDEs[J]. Commun. Math. Phys., 1999, 199:351-395.
[7]孔令华.一些非线性发展方程(组)的辛和多辛算法[D].合肥:中国科学技术大学, 2007.
[8] Bridges T J, Reich S. Numerical methods for Hamiltonian PDEs[J]. J. Phys. A:Math. Gen., 2006, 39:5287-5320.
[9] Reich S. Multi-symplectic Runge-Kutta collocation methods for Hamiltonian waveequations[J]. J. Comput. Phys., 2000, 157:473-499.
[10] Bridges T J, Reich S. Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations[J]. Physica D, 2001, 152-153:491-504.
[11] Chen J, Qin M. Multi-symplectic Fourier pseudospectral method for the nonlinearSchr?dinger equation[J]. Electr. Numer. Anal., 2001, 12:193-204.
[12] Wang J. On multisymplecticity of Sinc-Gauss-Legendre collocation discretizationfor some Hamiltonian PDEs[J]. Appl. Math. Comput., 2008, 199:105-121.
[13] Hong J, Liu X, Li C. Multi-symplectic Runge-Kutta-Nystr?m methods for nonlin-ear Schr?dinger equations with variable coefficients[J]. J. Comput. Phys., 2007,226:1968-1984.
[14] Moore B, Reich S. Backward error analysis for multi-symplectic integration meth-ods[J]. Numer. Math., 2003, 95:625-652.
[15] Cai J. A multisymplectic explicit scheme for the modified regularized long-waveequation[J]. J. Comput. Appl. Math., 2010, 234:899-905.
[16] Ryland B N, McLachlan B I, Frank J. On multisymplecticity of partitioned Runge-Kutta and splitting methods[J]. Inter. J. Comput. Math., 2007, 84:847-869.
[17] Wang Y, Qin M. Multisymplectic schemes for the nonlinear Klein-Gordon equa-tion[J]. Mathematical and Computer Modelling, 2002, 36:963-977.
[18] Wang Y, Wang B. High-order multi-symplectic schemes for the nonlinear Klein-Gordon equation[J]. Appl. Math. Comput., 2005, 166:608-632.
[19] Chen J. Symplectic and multisymplectic Fourier pseudospectral discretizations forthe Klein-Gordon equation[J]. Lett. Math. Phys., 2006, 75:293-305.
[20] Chen J, Qin M, Tang Y. Symplectic and multi-symplectic methods for the nonlinearSchr?dinger equation[J]. Comput. Math. Applic., 2002, 43:1095-1106.
[21] Hong J, Liu Y. A novel numerical approach to simulating nonlinear Schr?dingerequations with varying coefficients[J]. Appl. Math. Lett., 2003, 16:59-765.
[22] Hong J, Liu Y, Munthe-Kaas H,等. Globally conservative properties and error esti-mation of a multi-symplectic scheme for Schr?dinger equations with variable coef-ficients[J]. Appl. Numer. Math., 2006, 56:814-843.
[23] Cai J. Multisymplectic schemes for strongly coupled Schr?dinger system[J]. Appl.Math. Comput., 2010, 216:2417-2429.
[24] Hong J, Kong L. Novel multi-symplectic integrators for nonlinear fourth-orderSchr?dinger eqaution with trapped term[J]. Commun. Comput. Phys., 2010, 7:613-630.
[25]赵平福. KdV方程保结构计算方法的研究[D].北京:中国科学院数学与系统科学研究院, 2000.
[26] Wang Y, Wang B, Qin M. Numerical implementation of the multisymplectic Preiss-man scheme and its equivalent schemes[J]. Appl. Math. Comput., 2004, 149:299-326.
[27] Zhao P, Qin M. Multisymplectic geometry and multisymplectic Preissman schemefor the KdV equation[J]. J. Phys. A: Math. Gen., 2000, 33:3613-3626.
[28] Asher U M, Mclachlan R I. Multisymplectic box shemes and the Korteweg-de Vriesequation[J]. Appl. Numer. Math., 2004, 48:255-269.
[29] Asher U M, Mclachlan R I. On symplectic and multisymplectic shemes for the KdVequation[J]. Journal of Scientific Computing, 2005, 25:83-104.
[30]胡伟鹏,邓子辰,李文成. KdV方程的多辛算法及其孤立子解的数值模拟[J].西北工业大学, 2008, 26(2):128-131.
[31] Chen J. Multisymplectic Geometry, Local Conservation Laws and a Multisymplec-tic Integrator for the Zakharov–Kuznetsov Equation[J]. Lett. Math. Phys., 2003,63:115-124.
[32] Liu T, Qin M. Multisymplectic geometry and multisymplectic Preissman scheme forthe KP equation[J]. J. Math. Phys., 2002, 43:4060-4077.
[33] Hong J, Jiang S, Li C. Explicit multi-symplectic methods for Klein-Gordon-Schr?dinger equations[J]. J. Comput. Phys., 2009, 228:3517-3532.
[34] Kong L, Zhang J, Cao Y,等. Semi-explicit symplectic partitoned Runge-KuttaFourier pseudo-spectral scheme for Klein-Gordon-Schr?dinger equations[J]. Com-put. Phys. Commun., 2010, 181:1369-1377.
[35]孔令华.对称正则长波方程的辛算法[D].福州:华侨大学, 2004.
[36]曾文平,孔令华,单双荣. SRLW方程的多辛中点格式[J].厦门大学学报(自然科学版), 2006, 45(2):168-170.
[37] CaiJ. Multisymplecticnumericalmethodsfortheregularizedlong-waveequation[J].Comput. Phys. Commun., 2009, 180:1821-1831.
[38]孔令华,曾文平,刘儒勋,等. SRLW方程的多辛格式及其守恒律[J].中国科学技术大学学报, 2005, 35(6):770-776.
[39] Kong L, Hong J, Zang J. Splitting multi-symplectic integrators for Maxwell's equa-tion[J]. J. Comput. Phys., 2010, 229:4259-4278.
[40]曾文平,黄浪扬,秦孟兆.“Good”Boussinesq方程的多辛算法[J].应用数学和力学, 2002, 23(7):743-747.
[41]黄浪扬.“Good”Boussinesq方程的多辛Fourier拟谱算法[J].华侨大学学报(自然科学版), 2007, 28(1):92-95.
[42]黄浪扬.“good”Boussinesq方程多辛格式的构造[D].福州:华侨大学, 2004.
[43]王雨顺,王斌,季仲贞.孤立波方程的保结构算法[J].计算物理, 2004, 21(5):386-400.
[44]陈景波.无穷维Hamilton系统保结构算法一些问题的研究[D].北京:中国科学院数学与系统科学研究院, 2001.
[45] Chen J. Multisymplectic pseudospectral discretizations for (3+1)-dimensionalKlein–Gordon equation[J]. Commun. Theoretical Phys., 2008, 50(5):1052-1054.
[46]郭柏灵,田立新,杨灵娥,等. Camassa-Holm方程[M].北京:科学出版社, 2008.
[47] ChiuP,LeeL,SheuTWH. Asixth-orderdualpreservingalgorithmfortheCamassa-Holm equation[J]. J. Comput. Appl. Math., 2010, 233:2767-2778.
[48] Kalisch H, Lenells J. Numerical study of traveling-wave solutions for the Camassa-Holm equation[J]. Chaos Soltions and Fractrals, 2005, 25:287-297.
[49] Lenells J. Traveling Wave Solutions of the Camassa-Holm and Korteweg-de VriesEquations[J]. J. Nonlinear Math. Phys., 2008, 11(4):508-520.
[50] Henry D. Compactly Supported Solutions of the Camassa-Holm Equation[J]. J.Nonlinear Math. Phys., 2005, 12(3):342-347.
[51] Cohen D, Owren B, Raynaud X. Multi-symplectic integration of the Camassa-Holmequation[J]. J. Comput. Phys., 2008, 227:5492-5512.
[52] Ayd?n A, Karas?zen B. Multisymplectic box schemes for the complex modifiedKorteweg-de Vries equation[J]. J. Math. Phys., 2010, 51:083511-083534.
[53] Ito M. Symmetries and conservation laws of a coupled nonlinear wave equation[J].Phys. Lett. A, 1982, 91:405-420.
[54] Guha-Roy C. Solution of coupled KdV-Type Equations[J]. Intern. J. TheoreticalPhys., 1990, 29(8):863-866.
[55] Xu Y, Shu C.-W. Local discontinuous Galerkin methods for the Kuramoto-SivashinskyequationsandtheIto-typecoupledKdVequations[J]. Comput.MethodsAppl. Mech. Engrg., 2006, 195:3430-3447.
[56] Li P, Liu R. Numerical study on generalized KdV equations and Ito-type cou-pled KdV equations[J]. Journal of University Science Technology of China, 2008,38(9):1042-1067.
[57] IsmailMS,TahaTR. NumericalsimulationofcouplednonlinearSchr?dingerequa-tion[J]. Math. Comput. Simul., 2001, 56:547-562.
[58] Xu Y, Shu C.-W. Local discontinuous Galerkin methods for nonlinear Schr?dingerequaitons[J]. J. Comput. Phys., 2005, 205:72-97.
[59] Sun J, Qin M. Multi-symplectic methods for the coupled 1D nonlinear Schr?dingersystem[J]. Comput. Phys. Commun., 2003, 155:221-235.
[60] Ayd?n A, Karas?zen B. Symplectic and multi-symplectic methods for coupled non-linear Schr?dinger equations with periodic solutions[J]. Comput. Phys. Commun.,2007, 117:566-583.
[61] Aydln A. Multi-symplectic integration of N-coupled nonlinear Schr(o|¨)dinger equa-tions with destabilized periodic wave solutions[J]. Chaos Solitons and Fractrals,2009, 41:735-751.
[62] Ismail M S. A fourth-order explicit schemes for the coupled nonlinear Schr?dingerequaiton[J]. Appl. Math. Comput., 2008, 196:273-284.
[63] Zakharov V, Synakh V. The nature of self-focusing singularity[J]. Soviet Physics,JEPT, 1975, 41:465-468.
[64] Glassey R. On the blowing up of solutions to the Cauchy problem for nonlinearSchr?dinger equations[J]. J. Math. Phys., 1977, 18:1794-1797.
[65] Sulem P, Sulem C, Patera A. Numerical simulation of singular solutions to the two-dimensional cubic Schr?dinger equation[J]. Communications on Pure and AppliedMathematics, 1984, 37:755-778.
[66] Zakharov V, Kuznetsov E. Three-dimensional soliton[J]. Soviet Physics, JEPT,1974, 39:285-286.
[67] Iwasaki H, Toh S, Kawahara T. Cylindrical quasi-solutions of the Zakharov-Kuznetsov equation[J]. Physica D, 1990, 43:293-303.
[68] Xu Y, Shu C.-W. Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations[J]. Phys. D, 2005, 208:21-58.
[69]蒋斌. 2+1维孤立波方程的多辛格式[D].南京:南京师范大学, 2007.
[70] Hammack J, Schefner N, Segur H. Two-dimensional periodic waves in shallowwater[J]. Journal of Fluid Mechanics, 1989, 209:567-589.
[71] Hammack J, McCallister D, Schefner N,等. Two-dimensional periodic waves inshallow water. II. Asymmetric waves[J]. Journal of Fluid Mechanics, 1995, 285:95-122.
[72] Katsis C, Akylas T R. On the excitation of long nonlinear water waves by a movingpressure distribution. Part 2: Three-dimensional effects[J]. J. Fluid. Mech., 1987,177:49-65.
[73] Feng B F, Mitsui T. A finite differnce method for the Korteweg-de Vries and theKadomtsev-Petviashvili equations[J]. J. Comput. Appl. Math., 1998, 90:95-116.
[74] Wineberg S B, Mcgrath J F, Gabl E F,等. Implicit spectral methods for wave prop-agation problem[J]. J. Comput. Phys., 1991, 97:311-336.
[75] Feng B F, Kawahara T, Mitsui T. A conservative spectral method for several two-dimensional nonlinear wave equations[J]. J. Comput. Phys., 1999, 153:467-487.
[2]冯康,秦孟兆.哈密尔顿系统的辛几何算法[M].杭州:浙江科学技术出版社,2002.
[3] FengK,QinM. SymplecticgeometricalgorithmsforHamiltoniansystems[M]. NewYork: Springer-Verlag, 2009.
[4] Bridges T J. Multi-symplectic structures and wave propagation[J]. Math. Proc.Camb. Phil. Soc., 1997, 121:147-190.
[5] Bridges T J, Reich S. Multi-symplectic integrators: numerical schemes for Hamil-tonian PDEs that conserve symplecticity[J]. Phys. Lett. A, 2001, 284:184-193.
[6] Marsden J E, Patrick G W, Shkoller S. Multisymplectic geometry, variational inte-grators, and nonlinear PDEs[J]. Commun. Math. Phys., 1999, 199:351-395.
[7]孔令华.一些非线性发展方程(组)的辛和多辛算法[D].合肥:中国科学技术大学, 2007.
[8] Bridges T J, Reich S. Numerical methods for Hamiltonian PDEs[J]. J. Phys. A:Math. Gen., 2006, 39:5287-5320.
[9] Reich S. Multi-symplectic Runge-Kutta collocation methods for Hamiltonian waveequations[J]. J. Comput. Phys., 2000, 157:473-499.
[10] Bridges T J, Reich S. Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations[J]. Physica D, 2001, 152-153:491-504.
[11] Chen J, Qin M. Multi-symplectic Fourier pseudospectral method for the nonlinearSchr?dinger equation[J]. Electr. Numer. Anal., 2001, 12:193-204.
[12] Wang J. On multisymplecticity of Sinc-Gauss-Legendre collocation discretizationfor some Hamiltonian PDEs[J]. Appl. Math. Comput., 2008, 199:105-121.
[13] Hong J, Liu X, Li C. Multi-symplectic Runge-Kutta-Nystr?m methods for nonlin-ear Schr?dinger equations with variable coefficients[J]. J. Comput. Phys., 2007,226:1968-1984.
[14] Moore B, Reich S. Backward error analysis for multi-symplectic integration meth-ods[J]. Numer. Math., 2003, 95:625-652.
[15] Cai J. A multisymplectic explicit scheme for the modified regularized long-waveequation[J]. J. Comput. Appl. Math., 2010, 234:899-905.
[16] Ryland B N, McLachlan B I, Frank J. On multisymplecticity of partitioned Runge-Kutta and splitting methods[J]. Inter. J. Comput. Math., 2007, 84:847-869.
[17] Wang Y, Qin M. Multisymplectic schemes for the nonlinear Klein-Gordon equa-tion[J]. Mathematical and Computer Modelling, 2002, 36:963-977.
[18] Wang Y, Wang B. High-order multi-symplectic schemes for the nonlinear Klein-Gordon equation[J]. Appl. Math. Comput., 2005, 166:608-632.
[19] Chen J. Symplectic and multisymplectic Fourier pseudospectral discretizations forthe Klein-Gordon equation[J]. Lett. Math. Phys., 2006, 75:293-305.
[20] Chen J, Qin M, Tang Y. Symplectic and multi-symplectic methods for the nonlinearSchr?dinger equation[J]. Comput. Math. Applic., 2002, 43:1095-1106.
[21] Hong J, Liu Y. A novel numerical approach to simulating nonlinear Schr?dingerequations with varying coefficients[J]. Appl. Math. Lett., 2003, 16:59-765.
[22] Hong J, Liu Y, Munthe-Kaas H,等. Globally conservative properties and error esti-mation of a multi-symplectic scheme for Schr?dinger equations with variable coef-ficients[J]. Appl. Numer. Math., 2006, 56:814-843.
[23] Cai J. Multisymplectic schemes for strongly coupled Schr?dinger system[J]. Appl.Math. Comput., 2010, 216:2417-2429.
[24] Hong J, Kong L. Novel multi-symplectic integrators for nonlinear fourth-orderSchr?dinger eqaution with trapped term[J]. Commun. Comput. Phys., 2010, 7:613-630.
[25]赵平福. KdV方程保结构计算方法的研究[D].北京:中国科学院数学与系统科学研究院, 2000.
[26] Wang Y, Wang B, Qin M. Numerical implementation of the multisymplectic Preiss-man scheme and its equivalent schemes[J]. Appl. Math. Comput., 2004, 149:299-326.
[27] Zhao P, Qin M. Multisymplectic geometry and multisymplectic Preissman schemefor the KdV equation[J]. J. Phys. A: Math. Gen., 2000, 33:3613-3626.
[28] Asher U M, Mclachlan R I. Multisymplectic box shemes and the Korteweg-de Vriesequation[J]. Appl. Numer. Math., 2004, 48:255-269.
[29] Asher U M, Mclachlan R I. On symplectic and multisymplectic shemes for the KdVequation[J]. Journal of Scientific Computing, 2005, 25:83-104.
[30]胡伟鹏,邓子辰,李文成. KdV方程的多辛算法及其孤立子解的数值模拟[J].西北工业大学, 2008, 26(2):128-131.
[31] Chen J. Multisymplectic Geometry, Local Conservation Laws and a Multisymplec-tic Integrator for the Zakharov–Kuznetsov Equation[J]. Lett. Math. Phys., 2003,63:115-124.
[32] Liu T, Qin M. Multisymplectic geometry and multisymplectic Preissman scheme forthe KP equation[J]. J. Math. Phys., 2002, 43:4060-4077.
[33] Hong J, Jiang S, Li C. Explicit multi-symplectic methods for Klein-Gordon-Schr?dinger equations[J]. J. Comput. Phys., 2009, 228:3517-3532.
[34] Kong L, Zhang J, Cao Y,等. Semi-explicit symplectic partitoned Runge-KuttaFourier pseudo-spectral scheme for Klein-Gordon-Schr?dinger equations[J]. Com-put. Phys. Commun., 2010, 181:1369-1377.
[35]孔令华.对称正则长波方程的辛算法[D].福州:华侨大学, 2004.
[36]曾文平,孔令华,单双荣. SRLW方程的多辛中点格式[J].厦门大学学报(自然科学版), 2006, 45(2):168-170.
[37] CaiJ. Multisymplecticnumericalmethodsfortheregularizedlong-waveequation[J].Comput. Phys. Commun., 2009, 180:1821-1831.
[38]孔令华,曾文平,刘儒勋,等. SRLW方程的多辛格式及其守恒律[J].中国科学技术大学学报, 2005, 35(6):770-776.
[39] Kong L, Hong J, Zang J. Splitting multi-symplectic integrators for Maxwell's equa-tion[J]. J. Comput. Phys., 2010, 229:4259-4278.
[40]曾文平,黄浪扬,秦孟兆.“Good”Boussinesq方程的多辛算法[J].应用数学和力学, 2002, 23(7):743-747.
[41]黄浪扬.“Good”Boussinesq方程的多辛Fourier拟谱算法[J].华侨大学学报(自然科学版), 2007, 28(1):92-95.
[42]黄浪扬.“good”Boussinesq方程多辛格式的构造[D].福州:华侨大学, 2004.
[43]王雨顺,王斌,季仲贞.孤立波方程的保结构算法[J].计算物理, 2004, 21(5):386-400.
[44]陈景波.无穷维Hamilton系统保结构算法一些问题的研究[D].北京:中国科学院数学与系统科学研究院, 2001.
[45] Chen J. Multisymplectic pseudospectral discretizations for (3+1)-dimensionalKlein–Gordon equation[J]. Commun. Theoretical Phys., 2008, 50(5):1052-1054.
[46]郭柏灵,田立新,杨灵娥,等. Camassa-Holm方程[M].北京:科学出版社, 2008.
[47] ChiuP,LeeL,SheuTWH. Asixth-orderdualpreservingalgorithmfortheCamassa-Holm equation[J]. J. Comput. Appl. Math., 2010, 233:2767-2778.
[48] Kalisch H, Lenells J. Numerical study of traveling-wave solutions for the Camassa-Holm equation[J]. Chaos Soltions and Fractrals, 2005, 25:287-297.
[49] Lenells J. Traveling Wave Solutions of the Camassa-Holm and Korteweg-de VriesEquations[J]. J. Nonlinear Math. Phys., 2008, 11(4):508-520.
[50] Henry D. Compactly Supported Solutions of the Camassa-Holm Equation[J]. J.Nonlinear Math. Phys., 2005, 12(3):342-347.
[51] Cohen D, Owren B, Raynaud X. Multi-symplectic integration of the Camassa-Holmequation[J]. J. Comput. Phys., 2008, 227:5492-5512.
[52] Ayd?n A, Karas?zen B. Multisymplectic box schemes for the complex modifiedKorteweg-de Vries equation[J]. J. Math. Phys., 2010, 51:083511-083534.
[53] Ito M. Symmetries and conservation laws of a coupled nonlinear wave equation[J].Phys. Lett. A, 1982, 91:405-420.
[54] Guha-Roy C. Solution of coupled KdV-Type Equations[J]. Intern. J. TheoreticalPhys., 1990, 29(8):863-866.
[55] Xu Y, Shu C.-W. Local discontinuous Galerkin methods for the Kuramoto-SivashinskyequationsandtheIto-typecoupledKdVequations[J]. Comput.MethodsAppl. Mech. Engrg., 2006, 195:3430-3447.
[56] Li P, Liu R. Numerical study on generalized KdV equations and Ito-type cou-pled KdV equations[J]. Journal of University Science Technology of China, 2008,38(9):1042-1067.
[57] IsmailMS,TahaTR. NumericalsimulationofcouplednonlinearSchr?dingerequa-tion[J]. Math. Comput. Simul., 2001, 56:547-562.
[58] Xu Y, Shu C.-W. Local discontinuous Galerkin methods for nonlinear Schr?dingerequaitons[J]. J. Comput. Phys., 2005, 205:72-97.
[59] Sun J, Qin M. Multi-symplectic methods for the coupled 1D nonlinear Schr?dingersystem[J]. Comput. Phys. Commun., 2003, 155:221-235.
[60] Ayd?n A, Karas?zen B. Symplectic and multi-symplectic methods for coupled non-linear Schr?dinger equations with periodic solutions[J]. Comput. Phys. Commun.,2007, 117:566-583.
[61] Aydln A. Multi-symplectic integration of N-coupled nonlinear Schr(o|¨)dinger equa-tions with destabilized periodic wave solutions[J]. Chaos Solitons and Fractrals,2009, 41:735-751.
[62] Ismail M S. A fourth-order explicit schemes for the coupled nonlinear Schr?dingerequaiton[J]. Appl. Math. Comput., 2008, 196:273-284.
[63] Zakharov V, Synakh V. The nature of self-focusing singularity[J]. Soviet Physics,JEPT, 1975, 41:465-468.
[64] Glassey R. On the blowing up of solutions to the Cauchy problem for nonlinearSchr?dinger equations[J]. J. Math. Phys., 1977, 18:1794-1797.
[65] Sulem P, Sulem C, Patera A. Numerical simulation of singular solutions to the two-dimensional cubic Schr?dinger equation[J]. Communications on Pure and AppliedMathematics, 1984, 37:755-778.
[66] Zakharov V, Kuznetsov E. Three-dimensional soliton[J]. Soviet Physics, JEPT,1974, 39:285-286.
[67] Iwasaki H, Toh S, Kawahara T. Cylindrical quasi-solutions of the Zakharov-Kuznetsov equation[J]. Physica D, 1990, 43:293-303.
[68] Xu Y, Shu C.-W. Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations[J]. Phys. D, 2005, 208:21-58.
[69]蒋斌. 2+1维孤立波方程的多辛格式[D].南京:南京师范大学, 2007.
[70] Hammack J, Schefner N, Segur H. Two-dimensional periodic waves in shallowwater[J]. Journal of Fluid Mechanics, 1989, 209:567-589.
[71] Hammack J, McCallister D, Schefner N,等. Two-dimensional periodic waves inshallow water. II. Asymmetric waves[J]. Journal of Fluid Mechanics, 1995, 285:95-122.
[72] Katsis C, Akylas T R. On the excitation of long nonlinear water waves by a movingpressure distribution. Part 2: Three-dimensional effects[J]. J. Fluid. Mech., 1987,177:49-65.
[73] Feng B F, Mitsui T. A finite differnce method for the Korteweg-de Vries and theKadomtsev-Petviashvili equations[J]. J. Comput. Appl. Math., 1998, 90:95-116.
[74] Wineberg S B, Mcgrath J F, Gabl E F,等. Implicit spectral methods for wave prop-agation problem[J]. J. Comput. Phys., 1991, 97:311-336.
[75] Feng B F, Kawahara T, Mitsui T. A conservative spectral method for several two-dimensional nonlinear wave equations[J]. J. Comput. Phys., 1999, 153:467-487.
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