Please wait a minute...
Chin. Phys. B, 2010, Vol. 19(3): 030401    DOI: 10.1088/1674-1056/19/3/030401
GENERAL Prev   Next  

Approximate solution for the Klein-Gordon-Schr?dinger equation by the homotopy analysis method

Wang Jia(王佳)a), Li Biao(李彪) a)b)†, and Ye Wang-Chuan(叶望川)a)
a Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo 315211, China; b MM Key Lab, Chinese Academy of Sciences, Beijing 100080, China
Abstract  The Homotopy analysis method is applied to obtain the approximate solution of the Klein--Gordon--Schr?dinger equation. The Homotopy analysis solutions of the Klein--Gordon--Schr?dinger equation contain an auxiliary parameter which provides a convenient way to control the convergence region and rate of the series solutions. Through errors analysis and numerical simulation, we can see the approximate solution is very close to the exact solution.
Keywords:  Klein--Gordon--Schr?dinger equation      homotopy analysis method      approximate solution  
Received:  03 June 2009      Revised:  24 July 2009      Accepted manuscript online: 
PACS:  05.45.Yv (Solitons)  
  02.60.Lj (Ordinary and partial differential equations; boundary value problems)  
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 10735030), National Basic Research Program of China (Grant No.~2007CB814800), Ningbo Natural Science Foundation (Grant No.~2008A610017) and K.C. Wong Magna Fund in Ningbo University.

Cite this article: 

Wang Jia(王佳), Li Biao(李彪), and Ye Wang-Chuan(叶望川) Approximate solution for the Klein-Gordon-Schr?dinger equation by the homotopy analysis method 2010 Chin. Phys. B 19 030401

[1] Tsutsumim F 1978 J. Math. Anal. Appl. 66 358
[2] Dixon J M, Tuszynski J A and Clarkson P J 1997 FromNonlinearity to Coherence. Universal Features of Nonlinear Behaviorin Many-Body Physics (Cambridge: Cambridge University Press)
[2a] Gong L X 2006 Acta Phys. Sin. 55 4414 (in Chinese)
[2b] Deng Y X, Tu C H and Lü F Y 2009 Acta Phys. Sin. 58 3173 (in Chinese)
[2c] Chen X P, Lin J and Wang Z P 2007 Acta Phys. Sin. 56 3031 (in Chinese)
[3] Shen Y R 1984 Principles of Nonlinear Optics (New York:Wiley)
[4] Hasegawa A and Kodama Y 1995 Solitons in Optical Communications (New York: Oxford University Press)
[5] Karpman V I 1975 Nonlinear Waves in Dispersive Media (Oxford:Pergamon)
[6] Sulem C and Sulem P L 1999 The Nonlinear Schr?dingerEquation: Self-Focusing and Wave Collapse (Berlin: Springer)
[7] Petviashvili V and Pokhotelov O 1992 Solitary Waves in Plasmasand in the Atmosphere (Philadelphia: Gordon and Breach)
[8] Marklund M and Shukla P K 2006 Rev. Mod. Phys. 78 591
[9] Yu M Y and Shukla P K 1977 Plasma Phys. 19 889
[10] Wang M L and Zhou Y B 2003 Phys. Lett. A 318 84%11-15
[11] Miao C X and Xu G X 2006 J. Diff. Eqns. 227 365
[12] Ablowitz M J and Clarkson P A 1991 Solitons,Nonlinear Evolution Equations and Inverse Scattering (New York: CambridgeUniversity Press)
[13] Hirota R 1971 Phys. Rev. Lett. 27 1192
[14] Zha Q L and Li Z B 2008 Chin. Phys. B 17 2333
[15] Yang X D, Ruan H Y and Lou S Y 2008 Chin. Phys. Lett. 25 805%16-20
[16] Miurs M R 1978 B?cklund Transformation (Berlin: Springer)
[17] Weiss J, Tabor M and Carnevale G 1983 J. Math. Phys. 24 522
[18] Malfliet W 1992 Am. J. Phys. 60 650
[Malfliet W and Hereman W 1996 Phys. Scrpta 54 563
[19] Parkes E J and Duffy B R 1996 Comput. Phys. Commun. 98 288
[20] Wazwaz A M 2007 Math. Comput. Model. 45 473%21-25
[21] Fan E G 2000 Phys. Lett. A 277 212
[Fan E G 2002 Phys. Lett. A 294 26
[22] Yan Z Y 2001 Phys. Lett. A 292 100
[Yan Z Y and Zhang H Q 2001 Phys. Lett. A 285 355
[23] Chen C L, Zhang J and Li Y S 2007 Chin. Phys. 16 2167
[24] Chen Y, Li B and Zhang H Q 2003 Int. J. Mod. Phys. C 14 99
[25] Li B, Chen Y and Zhang H Q 2002 J.Phys. A: Math. Gen. 35 8253%26-30
[26] Mo J Q, Lin W T and Wang H 2007 Chin. Phys. 16 0951
[27] Wang M L 1996 Phys. Lett. A 213 279
[28] Zhang S F 2002 J. Phys. A: Math. Gen. 35 343
[29] Abbasbandy S and Darvishi M T 2005 Appl. Math. Comput. 163 1265
[30] Abbasbandy S and Darvishi M T 2005 Appl.Math. Comput. 17095%31-35
[31] Abbasbandy S 2006 Appl. Math. Comput. 173 493
[32] Liao S J 2003 Beyond Perturbation: Introduction to the Homotopy Analysis Method (Boca Raton: Chapman and Hall/CRC Press)
[33] Liao S J 1992 The Proposed Homotopy Analysis Techniques for the Solutionof Nonlinear Problems Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai (in English)
[34] Liao S J and Cheung K F 2003 J. Eng. Math. 45 105
[34a] Shi Y R, Wang Y H, Yang H J and Duan W S 2007 Acta Phys. Sin. 56 6791 (in Chinese)
[34b] Yang H J, Shi Y R, Duan W S and Lü K P 2007 ActaPhys. Sin. 56 3064 (in Chinese)
[35] Hayat T, Khan M and Asghar S 2004 Acta. Mech. 168 213
[36] Liao S J 2006 Commun. Nonlinear. Sci. Numer. Simulat. 11 326
[1] Influences of Marangoni convection and variable magnetic field on hybrid nanofluid thin-film flow past a stretching surface
Noor Wali Khan, Arshad Khan, Muhammad Usman, Taza Gul, Abir Mouldi, and Ameni Brahmia. Chin. Phys. B, 2022, 31(6): 064403.
[2] Application of asymptotic iteration method to a deformed well problem
Hakan Ciftci, H F Kisoglu. Chin. Phys. B, 2016, 25(3): 030201.
[3] Analytical approximate solution for nonlinear space-time fractional Klein–Gordon equation
Khaled A. Gepreel, Mohamed S. Mohamed. Chin. Phys. B, 2013, 22(1): 010201.
[4] Dynamic characteristics of resonant gyroscopes study based on the Mathieu equation approximate solution
Fan Shang-Chun(樊尚春), Li Yan(李艳), Guo Zhan-She(郭占社), Li Jing(李晶), and Zhuang Hai-Han(庄海涵) . Chin. Phys. B, 2012, 21(5): 050401.
[5] Approximate solution of the magneto-hydrodynamic flow over a nonlinear stretching sheet
Eerdunbuhe(额尔敦布和) and Temuerchaolu(特木尔朝鲁) . Chin. Phys. B, 2012, 21(3): 035201.
[6] Application of the homotopy analysis method for the Gross-Pitaevskii equation with a harmonic trap
Shi Yu-Ren (石玉仁), Liu Cong-Bo (刘丛波), Wang Guang-Hui (王光辉), Zhou Zhi-Gang (周志刚). Chin. Phys. B, 2012, 21(12): 120307.
[7] Approximate solutions of nonlinear PDEs by the invariant expansion
Wu Jiang-Long (吴江龙), Lou Sen-Yue (楼森岳). Chin. Phys. B, 2012, 21(12): 120204.
[8] Asymptotic solution for the Eiño time delay sea–air oscillator model
Mo Jia-Qi(莫嘉琪), Lin Wan-Tao(林万涛), and Lin Yi-Hua(林一骅). Chin. Phys. B, 2011, 20(7): 070205.
[9] Homotopic mapping solving method of the reduces equation for Kelvin waves
Mo Jia-Qi(莫嘉琪), Lin Yi-Hua(林一骅), and Lin Wan-Tao(林万涛). Chin. Phys. B, 2010, 19(3): 030202.
[10] Three types of generalized Kadomtsev-Petviashvili equations arising from baroclinic potential vorticity equation
Zhang Huan-Ping(张焕萍), Li Biao(李彪), Chen Yong (陈勇), and Huang Fei(黄菲). Chin. Phys. B, 2010, 19(2): 020201.
[11] A new method to obtain approximate symmetry of nonlinear evolution equation from perturbations
Zhang Zhi-Yong(张智勇), Yong Xue-Lin(雍雪林), and Chen Yu-Fu(陈玉福). Chin. Phys. B, 2009, 18(7): 2629-2633.
[12] Study on an extended Boussinesq equation
Chen Chun-Li(陈春丽), Zhang Jin E(张近), and Li Yi-Shen(李翊神). Chin. Phys. B, 2007, 16(8): 2167-2179.
[13] The sea--air oscillator model of decadal variations in subtropical cells and equatorial Pacific SST
Mo Jia-Qi(莫嘉琪), Lin Wan-Tao(林万涛), and Lin Yi-Hua(林一骅). Chin. Phys. B, 2007, 16(7): 1908-1911.
[14] Variational iteration method for solving perturbed mechanism of western boundary undercurrents in the Pacific
Mo Jia-Qi(莫嘉琪), Lin Wan-Tao(林万涛), and Wang Hui(王辉) . Chin. Phys. B, 2007, 16(4): 951-954.
No Suggested Reading articles found!