Please wait a minute...
Chin. Phys. B, 2017, Vol. 26(5): 050504    DOI: 10.1088/1674-1056/26/5/050504
GENERAL Prev   Next  

Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation

Zhen Wang(王振)1, Yupeng Qin(秦玉鹏)1, Li Zou(邹丽)2,3
1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;
2 School of Naval Architecture, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China;
3 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China
Abstract  We construct the Hirota bilinear form of the nonlocal Boussinesq (nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nlBq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nlBq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form. The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.
Keywords:  nonlocal Boussinesq equation      periodic wave solution      solitary waves      Riemann theta function  
Received:  05 January 2017      Revised:  22 January 2017      Published:  05 May 2017
PACS:  05.45.Yv (Solitons)  
  03.75.Lm (Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations)  
  02.30.Mv (Approximations and expansions)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 51579040, 51379033, and 51522902), the National Basic Research Program of China (Grant No. 2013CB036101), and Liaoning Natural Science Foundation, China (Grant No. 201602172).
Corresponding Authors:  Zhen Wang     E-mail:

Cite this article: 

Zhen Wang(王振), Yupeng Qin(秦玉鹏), Li Zou(邹丽) Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation 2017 Chin. Phys. B 26 050504

[1] Whitham G B 1974 Linear and Nonlinear Waves (New York: Wiley)
[2] Sachs A C 1988 Physica D 30 1
[3] Weiss J, M Tabor and G Carnevale 1983 J. Math. Phys. 24 522
[4] Hu X B, Li C X, Nimmo J J C and Yu G F 2005 J. Phys. A 38 195
[5] Zhang Y and Chen D Y 2005 Chaos, Solitons and Fractals 23 175
[6] Hitora R 2004 The Direct Method in Soliton Theory (Cambridge tracts in mathematics) (London: Cambridge University Press)
[7] Hitora R 1985 J. Phys. Soc. Jpn. 54 2409
[8] Hitora R, Hu X B and Tang X Y 2003 J. Math. Anal. Appl. 288 326
[9] Ablowitz M J and Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering (London: Cambridge University Press)
[10] Akhiezer N I 1990 Elements of the Theory of Ellipitic Functions (Providence: American Mathematical Society)
[11] Chen Y, Wang Q and Li B 2004 Z. Naturforsch. 59 529
[12] Zhou R G 1997 J. Math. Phys. 38 2535
[13] Lou S Y 1998 Z. Naturforsch. 53a 251
[14] Zhang D J, Wu H, Deng S F and Bi J B 2008 Commun. Theor. Phys. 49 1393
[15] Bell E T 1934 Ann. Math. 35 258
[16] Wang Z, Li D, Lu H and Zhang H 2005 Chin. Phys. B 14 2158
[17] Wang Z and Zhang H Q 2006 Chin. Phys. B 15 2210
[18] Novikov S P 1974 Funct. Anal. Appl. 8 236
[19] Dubrovin B A 1975 Funct. Anal. Appl. 9 215
[20] Its A R and Matveev V B 1975 Funct. Anal. Appl. 9 65
[21] Lax P D 1975 Commun. Pure Appl. Math. 28 141
[22] Mckean H P and Moerbeke P 1975 Invent. Math. 30 217
[23] Geng X G and Cao C W 2001 Nonlinearity 14 1433
[24] Geng X, Wu Y and Cao C 1999 J. Phys. A 32 3733
[25] Liu Q P, Hu X B and Zhang M X 2005 Nonlinearity 18 1597
[26] Tian B and Gao Y T 2001 Eur. Phys. J. B 22 351
[27] Hirota R and Satsuma J 1976 J. Phys. Soc. Jpn. 41 2141
[28] Yan Z Y 2010 Commun. Theor. Phys. 54 947
[29] Nakamura A 1979 J. Phys. Soc. Jpn. 47 1701
[30] Nakamura A 1980 J. Phys. Soc. Jpn. 48 1365
[31] Hu X B and Wang H Y 2006 Inverse Probl. 22 1903
[32] Fan E G and Hon Y C 2008 Phys. Rev. E 78 036607
[33] Fan E G 2009 J. Phys. A Math. Theor. 42 095206
[34] Ma W X and Zhou R G 2009 Mod. Phys. Lett. A 24 1677
[35] Tian S F and Zhang H Q 2014 Stud. Appl. Math. 132 212
[36] Wazwaz A M 2013 Appl. Math. Lett. 26 1094
[37] Lambert F, Loris I, Springael J and Willox R 1994 J. Phys. A 27 5325
[38] Willox R, Loris I and Springael J 1995 J. Phys. A 28 5963
[39] Loris I and Willox R 1996 J. Phys. Soc. Jpn. 65 383
[40] Farkas H M and Kra I 1992 Riemann Surfaces (New York: Springer-Verlag)
[41] Rauch H E and Lebowitz A 1973 Elloptic Functions, Theta Functions, and Riemann Surfaces (Baltitimore: William and Wilkins)
[1] Head-on collision between two solitary waves in a one-dimensional bead chain
Fu-Gang Wang(王扶刚), Yang-Yang Yang(杨阳阳), Juan-Fang Han(韩娟芳), Wen-Shan Duan(段文山). Chin. Phys. B, 2018, 27(4): 044501.
[2] Nucleus-acoustic solitary waves in self-gravitating degenerate quantum plasmas
D M S Zaman, M Amina, P R Dip, A A Mamun. Chin. Phys. B, 2018, 27(4): 040402.
[3] Solitary wave for a nonintegrable discrete nonlinear Schrödinger equation in nonlinear optical waveguide arrays
Li-Yuan Ma(马立媛), Jia-Liang Ji(季佳梁), Zong-Wei Xu(徐宗玮), Zuo-Nong Zhu(朱佐农). Chin. Phys. B, 2018, 27(3): 030201.
[4] Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method
Melis Zorsahin Gorgulu, Idris Dag, Dursun Irk. Chin. Phys. B, 2017, 26(8): 080202.
[5] (2+1)-dimensional dissipation nonlinear Schrödinger equation for envelope Rossby solitary waves and chirp effect
Jin-Yuan Li(李近元), Nian-Qiao Fang(方念乔), Ji Zhang(张吉), Yu-Long Xue(薛玉龙), Xue-Mu Wang(王雪木), Xiao-Bo Yuan(袁晓博). Chin. Phys. B, 2016, 25(4): 040202.
[6] A new model for algebraic Rossby solitary waves in rotation fluid and its solution
Chen Yao-Deng, Yang Hong-Wei, Gao Yu-Fang, Yin Bao-Shu, Feng Xing-Ru. Chin. Phys. B, 2015, 24(9): 090205.
[7] Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation
Reza Mohammadi. Chin. Phys. B, 2015, 24(5): 050206.
[8] Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation
Yin Jiu-Li, Xing Qian-Qian, Tian Li-Xin. Chin. Phys. B, 2014, 23(8): 080201.
[9] Exact solutions of the nonlinear differential—difference equations associated with the nonlinear electrical transmission line through a variable-coefficient discrete (G'/G)-expansion method
Saïdou Abdoulkary, Alidou Mohamadou, Ousmanou Dafounansou, Serge Yamigno Doka. Chin. Phys. B, 2014, 23(12): 120506.
[10] New exact solutions of (3+1)-dimensional Jimbo-Miwa system
Chen Yuan-Ming, Ma Song-Hua, Ma Zheng-Yi. Chin. Phys. B, 2013, 22(5): 050510.
[11] The extended cubic B-spline algorithm for a modified regularized long wave equation
İ. Dağ, D. Irk, M. Sarı. Chin. Phys. B, 2013, 22(4): 040207.
[12] Riemann theta function periodic wave solutions for the variable-coefficient mKdV equation
Zhang Yi, Cheng Zhi-Long, Hao Xiao-Hong. Chin. Phys. B, 2012, 21(12): 120203.
[13] Effects of dust size distribution on nonlinear waves in a dusty plasma
Chen Jian-Hong. Chin. Phys. B, 2009, 18(6): 2121-2128.
[14] New exact periodic solutions to (2+1)-dimensional dispersive long wave equations
Zhang Wen-Liang, Wu Guo-Jiang, Zhang Miao, Wang Jun-Mao, Han Jia-Hua. Chin. Phys. B, 2008, 17(4): 1156-1164.
[15] Applications of F-expansion method to the coupled KdV system
Li Bao-An, Wang Ming-Liang. Chin. Phys. B, 2005, 14(9): 1698-1706.
No Suggested Reading articles found!