Ahmet Bekir 1, Ömer Ünsal 2. (Received 5 September 2012, accepted 5 March 2013)

Transcription

1 ISSN print, online International Journal of Nonlinear Science Vol503 No,pp99-0 Exact Solutions for a Class of Nonlinear Wave Equations By Using First Integral Method Ahmet Bekir, Ömer Ünsal Eskişehir Osmangazi University, Art-Science Faculty, Department of Mathematics - Computer, Eskişehir-TÜRKİYE Eskişehir Osmangazi University, Art-Science Faculty, Department of Mathematics - Computer, Eskişehir-TÜRKİYE Received 5 September 0, accepted 5 March 03 Abstract:In this paper, we established the first integral method with symbolic computation to construct periodic,exponential, solitary wave solutions of some nonlinear wave equations With the aid of Maple, new exact solutions of four nonlinear equations in physics, namely, Duffing equation, Landau-Ginzburg-Higgs equation, Sine-gordon equation and coupled integrable dispersionless equations Keywords: Exact solutions, first integral method, duffing equations, Landau-Ginzburg-Higgs equation, Sinegordon equation, coupled integrable dispersionless equations Introduction Analytic solutions, in particular soliton solutions, of the nonlinear partial differential equations NPDEs, may well describe some phenomena in fluid dynamics, plasma physics, field theory, optics, solid-state physics, biophysics and so on As opposed to the linear problems, it is more difficult to solve the NPDEs In recent years, with the development of soliton theory, some methods for obtaining analytic solution to NPDEs have been proposed, such as the inverse scattering transformation [, ], Hirota bilinear tranformation [3 5], the tanh-sech method [6 8], extended tanh method [9, 0], sine-cosine method [, ], homogeneous balance method [3, 4], Exp-function method [5, 6] and G G -expansion method [7, 8], trial method [9, 0] Among those approaches, the first integral method is a tool to generate the soliton solutions for the NPDEs It is shown that an advantage of the first integral method lies in that the iterative algorithm is purely algebraic and computerizable by use of symbolic computation, while symbolic computation can be found in Refs [ 4] The rest of this paper is arranged as follows In Section, we simply provide the mathematical framework of the first-integral method In Section 3-6, in order to illustrate the method, four nonlinear equations are investigated, and abundant exact solutions are obtained which include new soliton-like solutions and trigonometric function solutions Finally, some conclusions are provided The first integral method The pioneer work Feng [] introduced the first integral method for a reliable treatment of the nonlinear PDEs The useful first integral method is widely used by some authors such as in [5 9] and by the reference therein Raslan has summarized for using first integral method [30] Step Consider a general nonlinear wave equation in the form P u, u t, u x, u xx, u tt, u xt, u xxx, = 0 Using a wave variable ξ = x ct We can rewrite Eq in the following nonlinear ODE QU, U, U, U, = 0 Corresponding author address: abekir@oguedutr Copyright c World Academic Press, World Academic Union IJNS03045/705

2 00 International Journal of Nonlinear Science, Vol503, No, pp 99-0 where the prime denotes the derivation with respect to ξ Equation is then integrated as long as all terms contain derivatives where integration constants are considered zeros Step Suppose that the solution of ODE can be written as follows: ux, t = fξ 3 Step 3 We introduce a new independent variable Xξ = fξ, Y = f ξ ξ, 4 which leads a system of X ξ ξ = Y ξ, Y ξ ξ = F Xξ, Y ξ 5 Step 4 By the qualitative theory of ordinary differential equations [3], if we can find the integrals to 5 under the same conditions, then the general solutions to 5 can be solved directly However, in general, it is really difficult for us to realize this even for one first integral, because for a given plane autonomous system, there is no systematic theory that can tell us how to find its first integrals, nor is there a logical way for telling us what these first integrals are We will apply the Division Theorem to obtain one first integral to 5 which reduces to a first order integrable ordinary differential equation An exact solution to is then obtained by solving this equation Now, let us recall the Division Theorem: Division Theorem: Suppose that P w, z, Qw, z are polynomials in Cw, z and P w, z is irreducible in Cw, z If Qw, z vanishes at all zero points of P w, z, then there exists a polynomial Gw, z in Cw, z such that Q[w, z] = P [w, z]g[w, z] 6 3 The Duffing equation Let us first consider the Duffing equation u tt + bu + cu 3 = 0 3 where b and c are real constants The Duffing equation describes the motion of a classical particle in a double well potential This equation can display chaotic behavior For c > 0, the equation represents a hard spring, and for c < 0, it represents a soft spring If c < 0, the phase portrait curves are closed Using the transformation Ux, t = fξ, ξ = x λt the Eq3 is carried to a ODE λ f ξ + bfξ + cf 3 ξ = 0 3 where the prime denotes the derivation with respect to ξ Using 4 we get Xξ = Y ξ 33 Y ξ = bxξ λ cx3 ξ λ 34 According to the first integral method, we suppose that Xξ and Y ξ are nontrivial solutions of 33, 34, and qx, Y = m a i XY i is an irreducible polynomial in the complex domain C[X, Y ] such that q[xξ, Y ξ] = m a i XY i = 0, 35 where a i X, i = 0,,, m are polynomials of X and a m X 0 Equation 35 is called the first integral to 33-34, due to the Division Theorem, there exists a polynomial gx + hxy in the complex domain C[X, Y ] such that dq dξ = q X X ξ + q m Y Y ξ = [gx + hxy ] a i XY i 36 IJNS for contribution: editor@nonlinearscienceorguk

3 Ahmet Bekir, Ömer Ünsal: Exact Solutions for a Class of Nonlinear Wave Equations By 0 In this example, we take two different cases, assuming that m = and m = in Eq35 Case I : Suppose that m =, by equating the coefficients of Y i i = 0,, on both sides of equation 36, we have a X = hxa X, 37 a 0 X = gxa X + hxa 0 X, 38 a XẎ = gxa bxλ cx3 0X = a X λ 39 Since a i X i = 0, are polynomials, then from 37 we deduce that a X is constant and hx = 0 For simplicity, take a X = Balancing the degrees of gx and a 0 X, we conclude that deggx = only Suppose that gx = A X + B 0, and A 0, then we find a 0 X a 0 X = A X + B 0 X + A 0 30 Substituting a 0 X, a X and gx in equation 39 and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain c A 0 = b λ, B c 0 = 0, A = λ 3 Using 3 into 35, we obtain Y ξ = b c λ c λ X ξ 3 Combining 3 with 33, we obtain the exact solution to 3 and then the exact solution to the Duffing equation can be written as: b bξ + Xξ = c tan C λ, 33 where C is integration constant Thus the periodic wave solution to the Duffing equation 3 can be written as: b bx λt + ux, t = c tan C λ 34 Case II : Suppose that m =, by equating the coefficients of Y i i = 0,,, 3 on both sides of equation 36, we have a 0 X = a X bx λ a X = hxa X, 35 a X = gxa X + hxa X, 36 cx3 λ + gxa X + hxa 0 X, 37 a XẎ = gxa 0X = a X bx λ cx3, 38 λ Since a X is a polynomial of X, then from 35 we deduce that a X is constant and hx = 0 For simplicity, take a X = Balancing the degrees of gx and a 0 X, we conclude that deggx = only Suppose that gx = A X + B 0,and A 0, then we find a X and a 0 X as a 0 X = a X = A X + B 0 X + A 0 39 A 8 + c λ X 4 + A B 0 X 3 b + λ + A A 0 + B 0 X + B 0 A 0 X + d 30 Substituting a 0 X, a X, a X and gx in equation 38 and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain A 0 = b c λc, A = c λ, B 0 = 0, d = b λ c 3 IJNS homepage:

4 0 International Journal of Nonlinear Science, Vol503, No, pp 99-0 Using 3 into 35, we obtain cb + cx ξ Y ξ = cλ Combining 3 with 33, we obtain the exact solution to 3 and then the exact solutions to the Duffing equation can be written as: b Xξ = i c tanh b λ ξ + C 33 3 where C is integration constant Thus the solitary wave solution to the Duffing equation 3 can be written as: b ux, t = i c tanh b λ x λt + C 34 Our results 34 can be compared with the result of Fuding et al [35] 4 The Landau-Ginzburg-Higgs equation Let us second consider the Landau-Ginzburg-Higgs equation u tt u xx m u + n u 3 = 0 4 where m and n are real constants This equation is studied in this paper based on the multi-symplectic theory in the Hamilton space Using the transformation Ux, t = fξ, ξ = x ct the Eq4 is carried to a ODE c f ξ f ξ m fξ + n f 3 ξ = 0 4 where the prime denotes the derivation with respect to ξ Using 4 we get Xξ = Y ξ 43 Y ξ = m Xξ c n X 3 ξ c According to the first integral method, we suppose that Xξ and Y ξ are nontrivial solutions of 43, 44, and qx, Y = m a i XY i is an irreducible polynomial in the complex domain C[X, Y ] such that q[xξ, Y ξ] = 44 m a i XY i = 0, 45 where a i X, i = 0,,, m are polynomials of X and a m X 0 Equation 45 is called the first integral to 43-44, due to the Division Theorem, there exists a polynomial gx + hxy in the complex domain C[X, Y ] such that dq dξ = q X X ξ + q m Y Y ξ = [gx + hxy ] a i XY i 46 In this example, we take two different cases, assuming that m = and m = in Eq45 Case I : Suppose that m =, by equating the coefficients of Y i i = 0,, on both sides of equation 46, we have a X = hxa X, 47 a 0 X = gxa X + hxa 0 X, 48 m a XẎ = gxa 0X = a X c n X 3 c 49 IJNS for contribution: editor@nonlinearscienceorguk

5 Ahmet Bekir, Ömer Ünsal: Exact Solutions for a Class of Nonlinear Wave Equations By 03 Since a i X, i = 0, are polynomials, then from 47 we deduce that a X is constant and hx = 0 For simplicity, take a X = Balancing the degrees of gx and a 0 X, we conclude that deggx = only Suppose that gx = A X + B 0, and A 0, then we find a 0 X a 0 X = A X + B 0 X + A 0 40 Substituting a 0 X, a X and gx in equation 39 and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain m A 0 =, B n c 0 = 0, A = n c 4 Using 4 into 45, we obtain m Y ξ = n c n c X ξ 4 Combining 4 with 43, we obtain the exact solution to 4 and then the exact solution to the Landau-Ginzburg- Higgs equation can be written as: Xξ = m e n c ξm+ c C m + c e ξm+, 43 c C m where C is integration constant Thus the travelling wave solution to the Landau-Ginzburg-Higgs equation 4 can be written as: c m e x ctm+ c C m + ux, t = c n e x ctm+ 44 c C m Case II : Suppose that m =, by equating the coefficients of Y i i = 0,,, 3 on both sides of equation 46, we have a X = hxa X, 45 a X = gxa X + hxa X, 46 a 0 X = a X m c n X 3 c + gxa X + hxa 0 X, 47 a XẎ = gxa 0X = a X m c n X 3 c, 48 Since a X is a polynomial of X, then from 45 we deduce that a X is constant and hx = 0 For simplicity, take a X = Balancing the degrees of gx and a 0 X, we conclude that deggx = only Suppose that gx = A X + B 0,and A 0, then we find a X and a 0 X as a 0 X = a X = A X + B 0 X + A 0 49 A 8 + n c X 4 + AB0X3 + m c + AA0 + B 0 X + B 0 A 0 X + d 40 Substituting a 0 X, a X, a X and gx in equation 48 and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain A 0 = m n c, m 4 A = n c, B 0 = 0, d = n c 4 IJNS homepage:

6 04 International Journal of Nonlinear Science, Vol503, No, pp 99-0 Using 4 into 45, we obtain Y ξ = n c nxξ + mnxξ m 4 Combining 4 with 43, we obtain the exact solution to 4 and then the exact solution to the Landau-Ginzburg- Higgs equation can be written as: m e C c m + Xξ = ux, t = n e ξ c m e C e c m ξ c m e x ct c m where C is integration constant Thus the travelling wave solution to the Landau-Ginzburg-Higgs equation 4 can be written as: C m e c m + n e e C c m x ct c m Our results 44 can be compared with the result of Fuding et al [35] 5 The Sine-Gordon equation Let us third consider the Sine-Gordon equation u tt u xx + u 6 u3 = 0 5 This equation appears in many scientific fields such as the propagation of fluxons in Josephson junctions between t- wo superconductors, the motion of rigid pendula attached to a stretched wire, solid state physics, nonlinear optics, and dislocations in metals Using the transformation Ux, t = fξ, ξ = x ct the Eq5 is carried to a ODE where the prime denotes the derivation with respect to ξ Using 4 we get c f ξ f ξ + fξ 6 f 3 ξ = 0 5 Xξ = Y ξ 53 Y ξ = Xξ c + X3 ξ 6c According to the first integral method, we suppose that Xξ and Y ξ are nontrivial solutions of 53, 54 and qx, Y = m a i XY i is an irreducible polynomial in the complex domain C[X, Y ] such that q[xξ, Y ξ] = 54 m a i XY i = 0, 55 where a i X,i = 0,,, m are polynomials of X and a m X 0 Equation 55 is called the first integral to 53-54, due to the Division Theorem, there exists a polynomial gx + hxy in the complex domain C[X, Y ] such that dq dξ = q X X ξ + q m Y Y ξ = [gx + hxy ] a i XY i 56 IJNS for contribution: editor@nonlinearscienceorguk

7 Ahmet Bekir, Ömer Ünsal: Exact Solutions for a Class of Nonlinear Wave Equations By 05 In this example, we take two different cases, assuming that m = and m = in Eq55 Case I : Suppose that m =, by equating the coefficients of Y i i = 0,, on both sides of equation 56, we have a X = hxa X, 57 a 0 X = gxa X + hxa 0 X, 58 a XẎ = gxa 0X = a X X c + X 3 6c 59 Since a i X i = 0, are polynomials, then from 57 we deduce that a X is constant and hx = 0 For simplicity, take a X = Balancing the degrees of gx and a 0 X, we conclude that deggx = only Suppose that gx = A X + B 0, and A 0, then we find a 0 X a 0 X = A X + B 0 X + A 0 50 Substituting a 0 X, a X and gx in equation 59 and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain 3 A 0 = c, B 0 = 0, A = 5 3c 3 Using 5 into 55, we obtain Y ξ = 3 c 3c 3 X ξ 5 Combining 5 with 53, we obtain the exact solution to 5 and then the exact solution to the Sine-Gordon equation can be written as: Xξ = 6 tanh c ξ + c C, 53 where C is integration constant Thus the solitary wave solution to the Sine-Gordon equation 5 can be written as: ux, t = 6 tanh c x ct + c C 54 Case II : Suppose that m =, by equating the coefficients of Y i i = 0,,, 3 on both sides of equation 56, we have a X = hxa X, 55 a X = gxa X + hxa X, 56 a 0 X = a X X c + X 3 6c + gxa X + hxa 0 X, 57 a XẎ = gxa 0X = a X X c + X 3 6c, 58 Since a X is a polynomial of X, then from 55 we deduce that a X is constant and hx = 0 For simplicity, take a X = Balancing the degrees of gx and a 0 X, we conclude that deggx = only Suppose that gx = A X + B 0,and A 0, then we find a X and a 0 X as a 0 X = a X = A X + B 0 X + A 0 59 A 8 c X 4 + A B 0 X 3 + c + A A 0 + B 0 X + B 0 A 0 X + d 50 Substituting a 0 X, a X, a X and gx in equation 58 and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain A 0 = c, A = 3c 3, B 3 0 = 0, d = c 5 IJNS homepage:

8 06 International Journal of Nonlinear Science, Vol503, No, pp 99-0 Using 5 into 55, we obtain Y ξ = 3 c X ξ 6 5 Combining 5 with 53, we obtain the exact solution to 5 and then the exact solutions to the Sine-Gordon equation can be written as: Xξ = i 6 tan c ξ + c C 53 where C is integration constant Thus the periodic wave solution to the Sine-Gordon equation 5 can be written as: ux, t = i 6 tan c x ct + c C 54 Comparing our results with Fuding s results [35] then it can be seen that the results are same 6 Coupled integrable dispersionless equations Let us consider coupled integrable dispersionless equations u xt + vw x = 0, 6 v xt vu x = 0, 6 w xt wu x = 0 63 This system physically describes a current-fed string interacting with an external magnetic field in three-dimensional Euclidean space It also appears geometrically as the parallel transport of each point of the curve along the direction of time where the connection is magnetic-valued Coupled integrable dispersionless equations are presented and solved by the inverse scattering method in [3] We use the transformation ux, t = uξ, vx, t = fξ, wx, t = wξ, ξ = x ct 64 where c is the wave speed and c 0 Substituting the Eqs64 into Eqs6-63 yields cu ξ fξwξ = 0 65 cf ξ + fξu ξ = 0 66 cw ξ + wξu ξ = 0 67 where the prime denotes the derivation with respect to ξ Integrating Eq65, we obtain u ξ = fξwξ + A 68 c where A is the integration constant Substituting Eq68 into Eqs66 and 67 yields f ξ + c f ξwξ + Bfξ = 0 69 w ξ + c fξw ξ + Bwξ = 0 60 where B = A/cEquations 69 and 60 are symmetrical about fξ and wξ Multiplying Eqs60 and 69 by fξ and wξ respectively and subtracting the latter from the former we have fξw ξ wξf ξ = 0 6 Equation 6 can also be written as fξw ξ wξf ξ = 0, 6 IJNS for contribution: editor@nonlinearscienceorguk

9 Ahmet Bekir, Ömer Ünsal: Exact Solutions for a Class of Nonlinear Wave Equations By 07 or so that wξ = 0, 63 fξ wξ = Dfξ 64 where D is the integration constant It is readily seen from Eq64 that wξ is proportional to fξ Substituting Eq64 into Eq69 or 60, we obtain that a second order nonlinear ordinary differential equation in fξ is given by [33] f ξ + Bfξ + D c f 3 ξ = 0 65 Using 4 we get Xξ = Y ξ 66 Y ξ = BXξ DX3 ξ c 67 According to the first integral method, we suppose that Xξ and Y ξ are nontrivial solutions of 66, 67, and qx, Y = m a i XY i is an irreducible polynomial in the complex domain C[X, Y ] such that q[xξ, Y ξ] = m a i XY i = 0, 68 where a i X,i = 0,,, m are polynomials of X and a m X 0 Equation 68 is called the first integral to 66-67, due to the Division Theorem, there exists a polynomial gx + hxy in the complex domain C[X, Y ] such that dq dξ = q X X ξ + q Y m Y ξ = [gx + hxy ] a i XY i 69 In this example, we take two different cases, assuming that m = and m = in Eq68 Case I : Suppose that m =, by equating the coefficients of Y i i = 0,, on both sides of equation 69, we have a X = hxa X, 60 a 0 X = gxa X + hxa 0 X, 6 a XẎ = gxa 0X = a X BX DX3 6 Since a i X i = 0, are polynomials, then from 60 we deduce that a X is constant and hx = 0 For simplicity, take a X = Balancing the degrees of gx and a 0 X, we conclude that deggx = only Suppose that gx = A X + B 0, and A 0, then we find a 0 X c a 0 X = A X + B 0 X + A 0 63 Substituting a 0 X, a X and gx in equation 6 and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain D A 0 = Bc, B D 0 = 0, A = c 64 Using 64 into 68, we obtain Y ξ = Bc D D X ξ 65 c IJNS homepage:

10 08 International Journal of Nonlinear Science, Vol503, No, pp 99-0 Combining 65 with 66, we obtain the exact solution to 65 and then the exact solution can be written as: B Xξ = c D tan B ξ + C, 66 uξ = BD wξ = c tan B ξ + C, 67 B c tan B ξ + C cbξ + C + Bcξ + C 68 where C and C are integration constants Thus the travelling wave solutions to the Coupled İntegrable Dispersionless equations 6,6,63 can be written as: B vx, t = c D tan B x ct + C 69 ux, t = BD wx, t = c tan B x ct + C, 630 B c tan B x ct + C cbx ct + C + Bcx ct + C 63 Case II : Suppose that m =, by equating the coefficients of Y i i = 0,,, 3 on both sides of equation 69, we have a X = hxa X, 63 a X = gxa X + hxa X, 633 a 0 X = a X BX DX3 + gxa X + hxa 0 X, 634 c a XẎ = gxa 0X = a X BX DX3, 635 Since a X is a polynomial of X, then from 63 we deduce that a X is constant and hx = 0 For simplicity, take a X = Balancing the degrees of gx and a 0 X, we conclude that deggx = only Suppose that gx = A X + B 0,and A 0, then we find a X and a 0 X as a 0 X = a X = A X + B 0 X + A D c X 0 X + B 0 A 0 X + d A 4 + A B 0 X 3 + B + A A Substituting a 0 X, a X, a X and gx in equation 635 and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain A 0 = cb D D, A = 4 D c, B 0 = 0, d = B c 4D 638 Using 638 into 68, we obtain D Bc + DX ξ Y ξ = 639 Dc Combining 639 with 66, we obtain the exact solution to 65 and then the exact solutions can be written as: B Xξ = c D tan B ξ + C 3, BD wξ = c tan B ξ + C 3, uξ = c B tan B ξ + C 3 + cb ξ + C 3 + Bcξ + C 4 c 640 IJNS for contribution: editor@nonlinearscienceorguk

11 Ahmet Bekir, Ömer Ünsal: Exact Solutions for a Class of Nonlinear Wave Equations By 09 where C 3 and C 4 are integration constants Thus the periodic wave solutions to the coupled integrable dispersionless equations 6,6,63 can be written as: B vx, t = c D tan B ct x + C 3, BD wx, t = c tan B ct x + C 3 ux, t = c B tan B ct x + C 3 + cb ct x + C 3 + Bcx ct + C 4 Comparing our results with Liu s results [33] then it can be seen that some of the obtained results are new, and the rest solutions are the same Remark With the aid of Maple, we have verified all solutions we obtained in Section, by putting them back into the original Eq 3, 4, 5 and 6 Remark Because the Landau-Ginburg-Higgs equation 4 is the first introduced and solved in this paper with the first integral method, all solutions are new and cannot be found in the literature Remark 3 Our results are different with that in the open literatures The obtained travelling wave solutions 64 can be converted into solitary wave solutions The obtained travelling wave solution 44 can be converted into periodic wave solution 64 7 Conclusions In this paper, the first integral method with the computerized symbolic computation system Maple is used for finding the solitary wave solutions, periodic wave solutions and exponential solutions to nonlinear equations arising in mathematical physics First integral method was employed to achieve the goal set for this work The validity of this method has been tested by applying it successfully to a class of nonlinear partial differential equations We foresee that our results can be found potentially useful for applications in mathematical physics and applied mathematics including numerical simulation Finally, it is worthwhile to mention that the method can also be applied to solve many other nonlinear evolution equations, nonlinear difference equations in mathematical physics and applied mathematics which will be investigated in our future work Acknowledgements This work was supported by Eskisehir Osmangazi University Scientific Research Projects Grant No: References [] Ablowitz, MJ, Clarkson, PA Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform Cambridge University Press, Cambridge, 990 [] Vakhnenko, V O, Parkes, E J, Morrison, A J A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation, Chaos, Solitons & Fractals, 74003:683 [3] Hirota R Direct method of finding exact solutions of nonlinear evolution equations, in: R Bullough, P Caudrey Eds, Backlund transformations Springer, Berlin, 980:57 [4] Wazwaz, AM The Hirota s direct method for multiple-soliton solutions for three model equations of shallow water waves Applied Mathematics and Computation, 0-008: 489 [5] Wazwaz, AM Multiple-soliton solutions for the KP equation by Hirota s bilinear method and by the tanh coth method Applied Mathematics and Computation, : 633 [6] Malfliet, W, Hereman, W The tanh method I: Exact solutions of nonlinear evolution and wave equations Physica Scripta, :563 [7] Wazwaz, AM The tanh method for travelling wave solutions of nonlinear equations Applied Mathematics and Computation, :73 IJNS homepage:

12 0 International Journal of Nonlinear Science, Vol503, No, pp 99-0 [8] Triki, H, Wazwaz, AM Sub-ODE method and soliton solutions for the variable-coefficient mkdv equation Applied Mathematics and Computation, 4009:370 [9] Fan, E Extended tanh-function method and its applications to nonlinear equations Phys Lett A, 77000: [0] Wazwaz, AM The extended tanh method for abundant solitary wave solutions of nonlinear wave equations Applied Mathematics and Computation, 87007:3 [] Wazwaz, AM A sine-cosine method for handling nonlinear wave equations Mathematical and Computer Modelling, :499 [] Bekir, A New solitons and periodic wave solutions for some nonlinear physical models by using sine-cosine method Physica Scripta, : 50 [3] Fan, E, Zhang H A note on the homogeneous balance method Phys Lett A, 46998: 403 [4] Abdel Rady, AS, Osman, ES, Khalfallah, M The homogeneous balance method and its application to the Benjamin Bona Mahoney BBM equation Applied Mathematics and Computation, 7400:385 [5] He, JH, Wu, XH Exp-function method for nonlinear wave equations Chaos, Solitons and Fractals, 30006:700 [6] Zhang, S Application of Exp-function method to a KdV equation with variable coefficients Phys Lett A, :448 [7] M Wang, X Li, J Zhang The G /G-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics Phys Lett A, 37008:47 [8] Bekir, A Application of the G /G-expansion method for nonlinear evolution equations Phys Lett A, 37008: 3400 [9] Liu, C S Trial equation method and its applications to nonlinear evolution equations Acta Physica Sinica, : 505 [0] Gurefe, Y, Sonmezoglu, A, Misirli, E Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics Pramana,, 770:03 [] Feng, Z S The first integral method to study the Burgers-KdV equationj Phys A: Math Gen, 3500:343 [] Feng, Z S, Wang, X H The first integral method to the two-dimensional Burgers-KdV equation Phys Lett A, :73 [3] Ahmed Ali, A H, Raslan, K R New solutions for some important partial differential equations Int J Nonlinear Science, 4 007:09 [4] Tascan, F, Bekir, A, Koparan, M Travelling wave solutions of nonlinear evolution equations by using the first integral method Communications in Nonlinear Science and Numerical Simulation, :80 [5] Taghizadeh, N, Mirzazadeh, M, Farahrooz, F Exact solutions of the nonlinear Schrödinger equation by the first integral method Journal of Mathematical Analysis and Applications, 3740:549 [6] Taghizadeh, N, Mirzazadeh, M The first integral method to some complex nonlinear partial differential equations Journal of Computational and Applied Mathematics, 3560:487 [7] Lu, B, Zhang, HQ, Fuding, X Travelling wave solutions of nonlinear partial equations by using the first integral method Applied Mathematics and Computation, 6400:39 [8] Deng, X Exact peaked wave solution of CH-γ equation by the first-integral method Applied Mathematics and Computation, : 806 [9] Abbasbandy, S, Shirzadi, A The first integral method for modified Benjamin Bona Mahony equation Communications in Nonlinear Science and Numerical Simulation, 5700:759 [30] Raslan, K R The first integral method for solving some important nonlinear partial differential equations Nonlinear Dynamics, : 8 [3] Ding, T R, Li, C Z Ordinary differential equations Peking University Press, Peking 996 [3] Konno, K, Oono, H New coupled integrable dispersionless equations J Phys Soc Jpn, : 377 [33] Liu, SK, Zhao, Q, Liu, SD The periodic solutions for coupled integrable dispersionless equations Chinese Physics B, 04 0: 0400 [34] Liu, SK, Liu, SD Nonlinear Equations in Physics Peking University Press, Peking 000 [35] Fuding, X, Xiaoshan, G Exact travelling wave solutions for a class of nonlinear partial differential equations Chaos, Solitons & Fractals, :3 [36] Yan, ZY New explicit and travelling wave solutions for a class of nonlinear evolution equations Acta Phys Sin, in Chinese : IJNS for contribution: editor@nonlinearscienceorguk