利用改進(jìn)的(G′/G)-展開法求廣義的(2+1)維Boussinesq方程的精確解

趙云梅,楊云杰,將艷

(1.紅河學(xué)院數(shù)學(xué)學(xué)院,云南 蒙自 661100;2.昆明學(xué)院數(shù)學(xué)系,云南 昆明 650031; 3.思茅師范高等?？茖W(xué)校數(shù)學(xué)系,云南 普洱 665000)

利用改進(jìn)的(G′/G)-展開法求廣義的(2+1)維Boussinesq方程的精確解

趙云梅1,楊云杰2,將艷3

(1.紅河學(xué)院數(shù)學(xué)學(xué)院,云南 蒙自 661100;2.昆明學(xué)院數(shù)學(xué)系,云南 昆明 650031; 3.思茅師范高等?？茖W(xué)校數(shù)學(xué)系,云南 普洱 665000)

利用改進(jìn)的(G′/G)-展開法,求廣義的(2+1)維Boussinesq方程的精確解,得到了該方程含有較多任意參數(shù)的用雙曲函數(shù)、三角函數(shù)和有理函數(shù)表示的精確解,當(dāng)雙曲函數(shù)表示的行波解中參數(shù)取特殊值時(shí),便得到廣義的(2+1)維Boussinesq方程的孤立波解.

廣義的(2+1)維Boussinesq方程;齊次平衡;改進(jìn)的(G′/G)-展開法;精確解

1 引言

本文研究下列廣義的(2+1)維Boussinesq方程[1]:

其中α,β,γ和δ是任意常數(shù),γδ/=0.

由于Boussinesq方程在物理學(xué)中有很好的應(yīng)用,故被許多學(xué)者用不同的方法研究,如文獻(xiàn)[1]用廣義的雅可比橢圓函數(shù)展開法研究了方程(1),獲得了許多行波解,在文獻(xiàn)[2]中,利用平面動(dòng)力系統(tǒng)分支理論,討論了各種參數(shù)狀態(tài)下行波解的存在性;當(dāng)α=β=γ=δ=1時(shí),文獻(xiàn)[3-5]對(duì)其解進(jìn)行了研究;當(dāng)α=C20,β=0時(shí),文獻(xiàn)[6]利用雅可比橢圓函數(shù)展開法對(duì)其解進(jìn)行研究;當(dāng)α=1,β=1,γ=-3,δ=1時(shí),文獻(xiàn)[7]利用廣田法對(duì)其解進(jìn)行了研究.

最近,由文獻(xiàn)[8]提出了(G′/G)-展開法,并成功地獲得了一些非線性演化方程的精確解.大量學(xué)者用該方法求解了大量的非線性偏微分方程(組)的精確解,之后出現(xiàn)了許多改進(jìn)的(G′/G)-展開法[9],廣義的(G′/G)-展開法[10]等,本文將利用改進(jìn)的G′/G-展開法研究廣義的(2+1)維Boussinesq方程,從而得到了方程(1)與目前現(xiàn)有文獻(xiàn)不同的新精確解.

2 廣義的(2+1)維Boussinesq方程的精確解

3 結(jié)論

本文利用改進(jìn)的(G′/G)-展開法,獲得了廣義的(2+1)維Boussinesq方程的用雙曲函數(shù)、三角函數(shù)和有理函數(shù)表示的精確解,其中當(dāng)雙曲函數(shù)表示的行波解中參數(shù)取特殊值時(shí),便得到廣義的(2+1)維Boussinesq方程的孤立波解,本文拓展了原來的(G′/G)-展開法,豐富了廣義的(2+1)維Boussinesq方程的解,從求解的過程可以看出,該方法在求解非線性微分方程精確解時(shí)具有更直接、更簡潔的特點(diǎn),此法也可以推廣到求其他非線性微分方程的精確解.

參考文獻(xiàn)

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[3] El-Sayed S M, Kaya D. The decomposition method for solving (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation[J]. Appl. Math. Comput, 2004,157(2):523-534.

[4] Abdel Rady A S, Osman E S, Khalfallah M. On soliton solutions of the (2+1) dimensional Boussinesq equation[J]. Appl. Math. Comput., 2009,(5):28.

[5] Chen Y, Yan Z Y, Zhang H Q. New explicit solitary wave solutions for (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation[J]. Phys. Lett. A., 2003,307(2/3):107-113.

[6] Liu S K, Fu Z T, Liu S D, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations[J]. Phys. Lett. A, 2001,289(1/2):69-74.

[7] Johnson R S. A two-dimensional Boussinesq equation for water waves and some of its solutions[J]. J. Fluid Mech., 1996,323:65-78.

[8] Wang M L.The(G′/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics[J]. Physics Letters A, 2008,372(4):417-423.

[9] Guo S M, Zhou Y B, Zhao C X. The improved (G′=G)-expansion method and its applications to the Broer- Kaup equations and approximate long water wave equations[J]. Applied Mathematics and Computation,2010, 216:1965-1971.

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Application of the improved(G′/G)-expansion method to exact solutions for generalized (2+1)-dimensional Boussinesq equation

Zhao Yunmei1,Yang Yun jie2,Jiang Yan3
(1. Department of Mathematics, Honghe University, Mengzi 66110, China; 2. Department of Mathematics, Kunming University, Kunming 650031, China; 3. Department of Mathematics, Simao Teachers′ College, Puer 665000, China)

In this paper,the im proved(G′/G)-expansion method is used to construct exact solutions of the generalized(2+1)-dimensional Boussinesq equation.As a resu lt,some new travelling wave solutions involving parameters,exp ressed by three typesof functionswhich are the hyperbolic functions,the trigonometric functions and the rational functions,are obtained.When the param eters are taken as special values,the solitary wave solutions are derived from the hyperbolic function solutions.

generalized(2+1)dimensional Boussinesq equation,homogeneous balance, im p roved(G′/G)-expansion method,exact solution

O175.2

A

1008-5513(2012)02-0176-05

2011-11-03.

國家自然科學(xué)基金(11161020);云南省科技廳項(xiàng)目(2011FZ193);昆明學(xué)院校級(jí)項(xiàng)目(XJ11L021).

趙云梅(1972-),碩士,副教授,研究方向：偏微分方程.

2010 MSC:35Q58,37K 50