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

doi:10.1088/1674-1056/26/5/050504

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
Accepted manuscript online:

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: wangzhen@dlut.edu.cn

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)

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

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