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1、Another New (G 0 G )-expansion Method and its Applications Houxia Shi, Rongfen Shi, Chao Li School of Mathematics and Statistics, Lan-zhou University, Lan-zhou 730000, China Abstract: In this paper, the (G 0 G )-expansion method is improved not only by making extension to the expression of the solut
2、ions of the NLEEs, but also by substituting the linear subsidiary equation in the G0 G -expansion method for a nonlinear one in form of F 02 = Pn i=0aiF i. Making use of the extended method, we study the compound KdV-Burgers equation, the Variant Boussinesq equations and the Hirota-Satsuma coupled K
3、dV equations and obtain rich new families of exact solutions, including the solution-like solutions, period-form solutions. PACS : 02.30.Jr, 04.20.Jb Keywords : extended (G 0 G)-expansion method, homogeneous balance principle, compound KdV-Burgers equation, Variant Boussinesq equations, Hirota-Satsu
4、ma coupled KdV equations. 1Introduction As mathematical models of the complex physics phenomena, the investigation of explicit solutions of the nonlinear evolution equations (NLEEs) will help one to understand these phenomena better. e-mail: shihouxia2008 1 Over last years, much work has been done b
5、y mathematicians and physicists to fi nd special solutions of NLEEs, such as the inverse scattering transform 1, the Backlund/Darboux transform 2-4, the Hirotas bilinear operators 5, the truncated Painleve expansion 6, the tanh-function expansion and its various extension 7-9, the Jacobi elliptic fu
6、nction expansion 10-11, the F-expansion 12-15, the sub-ODE method 16-19, the homogeneous balance method 20-22, the exp-function expansion method and so on. Recently, M.L. Wang, X.Zh Li and J.L. Zhang 24 present a (G 0 G)-expansion method to seek more travelling wave solutions of NLEEs. In this paper
7、, we develop the (G 0 G )-expansion method by changing the linear auxiliary equation of the (G 0 G )-expansion method into a nonlinear one in form of F 02 = Pn i=0aiF i, and we also make an extension to the expression of the solution. Then, we apply the extended method to the compound KdV-Burgers eq
8、uation, the Variant Boussinesq equations and the Hirota-Satsuma coupled KdV equations, and derive rich new families of exact solutions, including the solution-like solutions, period-form solutions. The rest of the paper is organized as follows. In Section 2, we describe the extended (G 0 G )-expansi
9、on method, and give the main steps of the method. In the subsequent sections, from Section 3 to Section 5, we illustrate the method in detail with the compound KdV-Burgers equation, Variant Boussinesq equations and Hirota-Satsuma coupled KdV equations. In Section 6, we briefl y make a summary to the
10、 extended method. 2 2Summary of the improved (G 0 G)-expansion method In this section, we describe the general theory of the improved (G 0 G)-expansion method. For a given NLEE, say, in two variables P(u,ut,ux,utt,uxx,uxt,) = 0,(2.1) where u = u(x,t) is an unknown function, P is a polynomial in u =
11、u(x,t) and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of the modifi ed (G 0 G)-expansion method. step 1. We suppose that u(x,t) = u(), = x V t,(2.2) the travelling wave variable (2.2) permits us re
12、ducing Eq.(2.1) to an ODE for u = u() P(u,V u0,u0,V 2u00,V u00,u00,) = 0. (2.3) Step 2. We assume that the solution of Eq.(2.3) can be expressed by a polynomial in (G 0 G) as follows: u() = 0+ N X i=N i(G 0 G )i,N6= 0,(2.4) where 0,i ,(i = N, ,N) are constants to be determined later, and G = G() sat
13、isfi es a nonlinear ODE in the form G 02 + qG2= p,p 6= 0,(2.5) 3 where q and p are constants, and N in Eq.(2.4) is a positive integer that can be determined by considering the homogeneous balance principle in Eq.(2.3). Step 3. Substituting Eq.(2.4) into Eq.(2.3) and using Eq.(2.5), the left-hand sid
14、e of Eq.(2.3) is converted into another polynomial in (G 0 G), collecting all terms with the same order of ( G0 G) i together. Equating each coeffi cient of the (G 0 G) i to zero, yields a set of algebraic equations for 0,i,(i = N, ,N) and V . Step 4. Assume that the constants 0,i,(i = N, ,N) and V
15、can be obtained by solving the algebraic equations in Step 3. Since the solutions of the Eq.(2.5) can be obtained, we substitute 0,i,(i = N, ,N),V and the general solutions of Eq.(2.5) into (2.4), then we have travelling wave solutions of Eq.(2.1). We fi nd that Eq.(2.5) admits many fundamental solu
16、tions which are listed as follows. Case 1. If q 0,p 0, say, the Eq.(2.5) is equivalent to G 02 + qG2= p,q 0,p 0,(2.6) then the Eq.(2.6) possesses three triangular type solutions G11() = rp q sin(q( c), G12() = rp q cos(q( c), G13() = 1 r p 2q sin(q( c) + 2 r p 2q cos(q( c), (2.7) where c is an arbitrary constant, and 1= 1,2= 1. Case 2. If q 0, the Eq.(2.5) is equivalent to G 02 qG2= p,
