利用广义变分迭代方法讨论了一类非线性强迫扰动Klein-Gordon方程.首先,用双曲函数待定系数法求得了无扰动方程孤子波.其次,利用泛函变分迭代原理得到了强迫扰动Klein-Gordon方程的一个摄动近似解.最后,论述了解的一致有效性.得到的近似解是解析式,它可对近似解进行解析运算,这对用简单的模拟方法得到的近似解是达不到的.
In this paper, a class of nonlinear forced disturbed Klein-Gordon equations are considered using the method of generalized variational iteration. Firstly, the solitary waves of an undisturbed Klein-Gordon equation are solved using the method of undetermined coefficients for hyperbolic functions. Then, perturbed approximate solutions for a soliton of a nonlinear forced disturbed Klein-Gordon equation are obtained using the functional variational iterative principle. Finally, the uniform validity for the approximate solutions is proved. The obtained approximate solution is an analytic expression. So it can be used for carrying out analytic operations. However, these cannot be obtained via a simple simulation.
[1] SALATHIEL Y, AMADOU Y, GARMBO B G, et al. Soliton solutions and traveling wave solutions for a discrete electrical lattice with nonlinear dispersion through the generalized Riccati equation mapping method[J]. Nonlinear Dynamics, 2017, 87(4):2435-2443.
[2] YU F J, LI L. Vector dark and bright soliton wave solutions and collisions for spin-1 Bose-Einstein condensate[J]. Nonlinear Dynamics, 2017, 87(4):2697-2713.
[3] CHOW B, LU P. On κ-noncollapsed complete noncompact shrinking gradient Ricci solitons which split at infinity[J]. Math Annalen, 2016, 366(3/4):1195-1206.
[4] SAZONOV S V. New type of extremely short vector solitons in a medium of asymmetric molecules[J]. JETP Letters, 2015, 102(12):834-840.
[5] 欧阳成, 姚静荪, 石兰芳, 等. 一类尘埃等离子体孤子波解[J]. 物理学报, 2014, 63(11):110203-110208.
[6] 石兰芳, 朱敏, 周先春, 等. 一类非线性发展方程孤立子行波解[J]. 物理学报,2014, 63(13):130201-130205.
[7] OUYANG C, CHENG L H, MO J Q. Solving a class of burning disturbed problem with shock layers[J]. Chin Phys B, 2012, 21(5):15-18.
[8] SHI L F, CHEN C S, ZHOU X C. The extended auxiliary equation method for the KdV equation with variable coefficients[J]. Chin Phys B, 2011, 20(10):803-811.
[9] 石兰芳, 周先春. 一类扰动Burgers方程的孤子同伦映射解[J]. 物理学报,2010, 59(5):2915-2918.
[10] MO J Q. Homotopic mapping solving method for gain fluency of a laser pulse amplifier[J]. Science in China G, 2009, 39(7):1007-1010.
[11] MO J Q, LIN S R. The homotopic mapping solution for the solitary wave for a generalized nonlinear evolution equation[J]. Chin Phys B, 2009, 18(9):3628-3631.
[12] MO J Q. Solution of travelling wave for nonlinear disturbed long-wave system[J]. Commun Theor Phys, 2011, 55(3):387-390.
[13] MO J Q, CHEN X F. Homotopic mapping method of solitary wave solutions for generalized complex Burgers equation[J]. Chin Phys B, 2010, 19(10):20-23.
[14] XU J Z, ZHOU Z F. Existence and uniqueness of anti-periodic solutions for a kind of nonlinear nth-order differential equation with multiple deviating arguments[J]. Ann Diff Eqn, 2012, 28(1):105-114.
[15] 徐建中, 周宗福. 一类四阶具有多个偏差变元p-Laplacian中立型微分方程周期解的存在性[J]. 重庆工商大学学报, 2012, 29(11):9-16.
[16] HE J H, WU X H. Construction of solitary solution and compaction-like solution by variational iteration method[J]. Chaos Solitons & Fractals, 2006, 29(1):108-113.
[17] ZHANG X A, CHEN K, DUAN Z I. Bound states of KG equation and Dirac for ring-shaped non-spherical oscillator scalar and vector potential[J]. Chin Phys, 2005, 14(1):42-44.
[18] ZHANG Q, YUE P, GONG L X. New exact solutions of nonlinear KG equation[J]. Chin Phys, 2006, 15(1):35-38.
[19] TEMAN R. Infinite-dimensional Dynamical System in Mechanics and Physics[M]. 2nd ed. New York:Springer, 1997.
[20] 张建文, 王旦霞, 吴润衡. 一类广义强阻尼Sine-Gordon方程的整体解[J]. 物理学报, 2008, 57(4):2021-2025.
[21] 韩兆秀. 非线性KG方程新的精确解[J]. 物理学报,2005, 54(4):1481-1484.
[22] 卢殿臣, 杨立波, 洪宝剑. 非线性KG方程新的精确解[J]. 江苏大学学报,2010, 31(1):120-124.
[23] CHANG K W, HOWES F A. Nonlinear Singular Perturbation Phenomena:Theory and Applications[M]. New York:Springer-Verlag, 1984.
[24] DE JAGER E M, JIANG F R. The Theory of Singular Perturbation[M]. Amsterdam:North-Holland Publishing Co, 1996.
[25] BARBU L, MOROSANU G. Singularly Perturbed Boundary-Value Problems[M]. Basel:Birkhauser, 2007.