Modified Reductive Perturbation Method as Applied to Long Water-Waves:The Korteweg-de Vries Hierarchy

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.6(28) No.,pp.-2 Modified Reductive Perturbation Method as Applied to Lon Water-Waves:The Kortewe-de Vries Hierarchy Hilmi Demiray Department of Mathematics,Faculty of Arts and Sciences Isik University, Sile 3498-Istanbul-Turkey (Received 6 July 27, accepted 4 June 28) Abstract:In this work, we extended the application of the modified reductive perturbation method to lon water waves and obtained the overnin equations of KdV hierarchy. Seekin a localized travellin wave solutions to these evolution equations we determined the scale parameter c so as to remove the possible secularities that miht occur. To indicate the power and eleance of the present method, we studied the same problem by use of the dressed solitary wave method and obtained exactly the same result. The present method is seen to be fairly simple as compared to the renormalization method of Kodama and Taniuti [4] and the multiple scale expansion method of Kraenkel et al [6]. Keywords:modified reductive perturbation method; water waves; Kortewe-deVries hierarchy Introduction In collisionless cold plasma, in fluid-filled elastic tubes and in shallow-water waves, due to nonlinearity of the overnin equations, for weakly dispersive case one obtains the Kortewe-de Vries (KdV) equation for the lowest order term in the perturbation expansion, the solution of which may be described by solitons (Davidson []). To study the hiher order terms in the perturbation expansion, the reductive perturbation method has been introduced by use of the stretched time and space variables (Taniuti [2]). However, in such an approach some secular terms appear which can be eliminated by introducin some slow scale variables (Suimoto and Kakutani [3]) or by a renormalization procedure of the velocity of the KdV soliton (Kodama and Taniuti [4]). Nevertheless, this approach remains somewhat artificial, since there is no reasonable connection between the smallness parameters of the stretched variables and the one used in the perturbation expansion of the field variables. The choice of the former parameter is based on the linear wave analysis of the concerned problem and the wave number or the frequency is taken as the perturbation parameter (Washimi and Taniuti [5]). On the other hand, at the lowest order, the amplitude and the width of the wave are expressed in terms of the unknown perturbed velocity, which is also used as the smallness parameter. This causes some ambiuity over the correction terms. Another attempt to remove such secularities is made by Kraenkel et al [6] for lon water waves by use of the multiple time scale expansion but could not obtain explicitly the correction terms to the wave speed. In order to remove these uncertainities, Malfliet and Wieers [7] presented a dressed solitary wave approach, which is based on the assumption that the field variables admit localized travellin wave solution. Then, for the lonwave limit, they expanded the field variables and the wave speed into a power series of the wave number, which is assumed to be the only smallness parameter, and obtained the explicit solution for various order terms in the expansion. However, this approach can only be used when one studies proressive wave solution to the oriinal nonlinear equations and it does not ive any idea about the form of evolution equations overnin the various order terms in the perturbation expansion. In our previous paper [8], we address: demiray@isikun.edu.tr Copyriht c World Academic Press, World Academic Union IJNS /57

2 2 International Journal of Nonlinear Science,Vol.6(28),No.,pp. -2 have presented a method so called the modified reductive perturbation method to examine the contributions of hiher order terms in the perturbation expansion and applied it to weakly dispersive ion-acoustic plasma waves and solitary waves in a fluid filled elastic tube [9]. In these works, we have shown that the lowest order term in the perturbation expansion is overned by the nonlinear Kortewe-de Vries equation, whereas the hiher order terms in the expansion are overned by the deenerate Kortewe-de Vries equation with non-homoeneous term. By employin the hyperbolic tanent method a proressive wave type of solution was souht and the possible secularities were removed by selectin the scalin parameter in a special way. The basic idea in this method was the inclusion of hiher order dispersive effects throuh the introduction of the scalin parameter c, to balance the hiher order nonlinearities with dispersion. The nelience of hiher order dispersive effects in the classical reductive perturbation method leads to the imbalance between the nonlinearity and the dispersion, which resulted in some secular terms in the solution of evolution equations. As a matter of fact, the renormalization method presented by Kodama and Taniuti [4] is different but rather involved formulation of the same idea. For further discussion of theses methods the reader is referred to the references [-3]. In the present work, we extended the application of the modified reductive perturbation method to lon water-waves and obtained the equations of KdV hierarchy. Seekin a localized proressive wave solutions to these equations we determined the scale parameter c so as to remove the possible secularities that miht occur. To indicate the power and eleance of the present method, we studied the same problem by use of the dressed solitary wave method and obtained exactly the same result. The present method is seen to be fairly simple as compared to the renormalization method of Kodama and Taniuti [4] and the multiple scale expansion method of Kraenkel et al [6]. 2 Modified reductive perturbation formalism for water waves We consider a two-dimensional incompressible inviscid fluid in a constant ravitational field. The space coordinates are denoted by (x, z) and the correspondin velocity components by (u, w). The ravitational acceleration is in neative z direction. The equations describin such a fluid are: u x + w =, (incompressibility) () u t + u u x + w u = p ρ x, (2) w t + u w x + w w = p, (3) ρ where ρ is the mass density and p is the fluid pressure function. Assumin that the flow is irrotational, the velocity vector can be derived from a scaler potential ˆφ(x, z, t) as u = ˆφ x, Then, the incompressibility condition reduces to w = ˆφ. (4) and the Euler equations become 2 ˆφ x ˆφ =, (5) 2 p p ρ = ˆφ t ˆφ [( 2 x )2 + ( ˆφ )2 ] z, (6) where p is an interation constant. We consider the case of a fluid of heiht h, bounded above by a steady atmosphere with pressure p. Let the upper surface be described by z = ˆψ(x, t). The kinematic boundary condition on this surface can be expressed as: ˆφ = ˆψ t + ˆφ x ˆψ x, on z = ˆψ(x, t). (7) IJNS for contribution: editor@nonlinearscience.or.uk

3 Hilmi Demiray: Modified Reductive Perturbation Method as Applied to 3 From equation (6), the dynamical boundary condition on this surface reads ˆφ t + ˆφ [( 2 x )2 + ( ˆφ )2 ] + ˆψ =, on z = ˆψ(x, t). (8) Finally, the lower boundary is supposed to be riid. Therefore, at z = h, the normal component of the velocity must vanish, i.e., ˆφ = at z = h. (9) Now, we shall consider the lonwave in shallow-water approximation to the above equations by applyin the modified reductive perturbation method developed by us (Demiray [8]). Accordin to this method, we introduce the followin coordinate stretchin ξ = ɛ /2 (x c t), τ = ɛ 3/2 ct, () where ɛ is a small parameter characterizin the smallness of certain physical entities, c and c are two constants to be determined from the solution. For our future purposes we introduce new variable as Introducin () and () into equations (5)-(9) we obtain φ = ɛ2 c ψ τ ɛc ψ ξ ˆφ = ɛ /2 φ, ˆψ = ɛψ. () ɛ 2 φ ξ φ =, (2) 2 φ ψ + ɛ2 ξ ξ at z = ɛψ, (3) ɛc φ τ c φ ξ + 2 [ɛ( φ ξ )2 + ( φ )2 ] + ψ =, at z = ɛψ, (4) and φ =, at z = h. (5) Now, we expand the functions φ and ψ, and the constant c into a suitable power series in the parameter ɛ : φ = φ + ɛφ + ɛ 2 φ 2 + ɛ 3 φ , ψ = ψ + ɛψ + ɛ 2 ψ 2 + ɛ 3 ψ , c = + ɛc + ɛ 2 c 2 + ɛ 3 c , (6) Introducin the expansions (6) into equations (2)-(5) the followin set of equations are obtained: O() equations O(ɛ) equations 2 φ =, (7) 2 φ =, at z = h. (8) φ φ z= =, c ξ z= + ψ =. (9) φ 2 φ φ =, (2) ξ2 = at z = h, φ ψ z= + c ξ =, [ φ τ c φ ξ + 2 ( φ ξ )2 ] z= + ψ =. (2) IJNS homepae:

4 4 International Journal of Nonlinear Science,Vol.6(28),No.,pp. -2 O(ɛ 2 ) equations φ 2 2 φ φ =, (22) ξ2 = at z = h, O(ɛ 3 ) equations [ φ 2 + ψ 2 φ 2 ] z= = ψ τ c ψ ξ + φ ψ ξ ξ z=, φ [c τ + φ τ c ( φ 2 ξ + ψ 2 φ ξ ) + φ ξ φ 3 φ ξ + 2 ( φ )2 ] z= + ψ 2 =. (23) 2 φ φ 2 =, (24) ξ2 =, at z = h, [ φ 3 + ψ 2 φ ψ 2 φ 2 φ ψ ξ ξ φ ψ ξ ξ ] ψ z= = c τ + ψ τ c ψ 2 ξ, φ [c 2 τ + c φ τ + φ 2 τ + ψ 2 φ τ c ( φ 3 ξ + ψ 2 φ 2 ξ + 3 φ 2 ψ2 2 ξ + ψ 2 φ ξ ) 2 ( φ ξ )2 + φ ξ ( φ 2 ξ + ψ 2 φ ξ ) + φ ( φ 2 + ψ 2 φ 2 ] z= + ψ 3 =. (25) 2. Solution of the field equations From the solution of the sets (8) and (9) we have φ = F (ξ, τ), ψ = c F ξ, (26) where F (ξ, τ) is an unknown function of its arument whose overnin equation will be obtained later. Similarly, from the solution of the equations (2) and (2) one obtains φ = 2 F 2 ξ 2 (z2 + 2hz) + G(ξ, τ), ψ = c G ξ [ F τ + 2 ( F ξ )2 ], c = (h) /2, (27) where G(ξ, τ) is another unknown function whose overnin equation will be obtained from the hiher order expansions. The solution of O(ɛ 2 ) equations, (22) and (23), yields the followin results φ 2 = 4 F 24 ξ 4 (z4 + 4hz 3 8h 3 z) 2 G 2 ξ 2 (z2 + 2hz) + H(ξ, τ), ψ 2 = c H ξ ( G τ + F G ξ ξ ) c F τ c h ψ 3 F ξ 3 h2 F 2 ( 2 ξ 2 )2, (28) where H(ξ, τ) is another unknown function whose overnin equation will be obtained later. The use of the boundary condition (23) 2 yields the followin evolution equation, which is the conventional KortewedeVries equation ψ τ + 3 ψ ψ 2 c ξ + h3 3 φ =. (29) 6c ξ3 As is seen from this equation, the lowest order term in the perturbation expansion is overned by the Kortewe-de Vries equation. IJNS for contribution: editor@nonlinearscience.or.uk

5 Hilmi Demiray: Modified Reductive Perturbation Method as Applied to 5 Usin the equation (29) in the equation (27), the function ψ can be expressed as ψ = c G ξ + ψ2 4h + h2 2 ψ 6 ξ 2. (3) In order to study the novelty of the present method we should study the hiher order expansion. For that purpose we shall ive the solution of O(ɛ 3 ) equations, which follows from (24)-(25) φ 3 = 6 F 72 ξ 6 (z6 +6hz 5 4h 3 z 3 +96h 5 z)+ 4 G 24 ξ 4 (z4 +4hz 3 8h 3 z) 2 H 2 ξ 2 (z2 +2hz)+I(ξ, τ), (3) where I(ξ, τ) is another unknown function whose overnin equation will be obtained from the hiher order expansions. The use of the boundary condition (25) 2 yields the followin evolution equation c 2h 5 5 Notin the relations 6 F ξ 6 + h3 4 G 3 ξ 4 + ψ 2 G ξ 2 + ψ 2 F ξ 2 + ψ G ξ ξ + F ψ ξ ξ + c ψ τ + ψ τ + 2 G ξ τ + c 2 F G ξ 2 ξ + c F 2 G ξ ξ 2 + c c F ξ = c ψ, the followin evolution equation is obtained where the non-homoeneous term S(ψ ) is defined by 2 F ξ τ + 4 F h2 ψ ξ 4 + ψ 2h2 ξ 3 F =. (32) ξ3 G ξ = ψ c 4c h ψ2 h2 2 ψ 6c ξ 2, (33) ψ τ + h3 3 ψ 6c ξ c ξ (ψ ψ ) = S(ψ ), (34) ξ S(ψ ) = 9h5 4 ψ 36c ξ 4 5h2 2 ψ ψ 2c ξ 2 3h2 ( ψ 48c ξ )2 + 8c h ψ3 + 3c ψ 2 + h3 c 2 ψ 4c 6c ξ 2. (35) Here, the equation (34) is the linearized Kortewe-deVries equation with non-homoeneous term S(ψ ) which contains the unknown coefficient c which is to be determined from the solution. 2.2 Travellin Wave Solution In this section we shall present a proressive wave solutions to equations (29) and (34). For that purpose we shall seek a solution to these equations in the followin form ψ = F (ζ), ψ = V (ζ), ζ = α(ξ βτ), (36) where α and β are two constants to be determined from the solutions. Introducin the proposed expression of ψ (ζ) into equation (29) we obtain βu + 3 UU + h3 α 2 U =, (37) 2 c 6c where a prime denotes the differentiation of the correspondin quantity with respect to ζ. In this work we shall be concerned with the localized travellin wave solution, i.e., U and its various order derivatives vanish as ζ. Interatin equation (37) with respect to ζ and usin the localization condition we obtain βu + 3 U 2 + h3 α 2 U =. (38) 4c 6c As is well-known this equation admits the proressive wave solution of the form U = asech 2 ζ, (39) IJNS homepae:

6 6 International Journal of Nonlinear Science,Vol.6(28),No.,pp. -2 where a is the amplitude of the solitary wave and the other quantities are defined by α = ( 3a 4h 3 )/2, β = a 2c. (4) Here, we note that the wave speed is proportional to the amplitude of the wave. To obtain the solution for V (ζ), we introduce (36) and (39) into equations (34) and (35), which yields βv + h3 6c α 2 V + 3 2c (UV ) = S(U). (4) Interatin this equation with respect to ζ and usin the localization condition, we obtain V + (2sech 2 ζ 4)V = 6a2 h sech6 ζ + 2a2 h sech4 ζ + (4ac 9a2 5h )sech2 ζ. (42) Here, we shall propose a solution for V of the followin form V = Asech 4 ζ + Bsech 2 ζ, (43) where A and B are two constants to be determined from the solution of (42). Carryin out the derivative of V we have V = 2Asech 6 ζ + (6A 6B)sech 4 ζ + 4Bsech 2 ζ (44) Insertin (43) and (44) into (42) and settin the coefficients of sech 6 ζ and sech 4 ζ equal to zero, one obtains A = 3a2 4h, and the remainin part of the equation (42) reads B = a2 2h, (45) (4c a 9a2 5h )sech2 ζ =. (46) If the constant c was equal to zero, as in the classical reductive perturbation method, the equation would not be balanced. In order to balance the equation the solution must contain a secular term. In order to remove the secular term, the equation (46) must be satisfied identically; from which we obtain c = 9a 2h. (47) Thus, in terms of real physical entities, the final solution takes the followin form with ψ = a sech 2 ζ + ɛ( 3a2 4h sech4 ζ + a2 2h sech2 ζ). (48) ζ = ɛ /2 {( 3a 4h 3 )/2 [x c ( + a 2h ɛ + 9a2 4h 2 ɛ2 )t]} (49) 3 Dressed solitary wave formalism for water waves In order to see the novelty of the present formulation, we shall study the same problem by use of the method so called the dressed solitary wave formalism. For this purpose, we shall examine the solution to equations (5), (7) - (9) of the followin form ˆφ = ˆφ(ξ, z), ˆψ = ˆψ(ξ), ξ = k(x V t), (5) where k is the wave number and V is the speed of the wave. Introducin (5) into equations (5), (7) -(9) we have the field equation k 2 2 ˆφ ξ ˆφ =, (5) 2 IJNS for contribution: editor@nonlinearscience.or.uk

7 Hilmi Demiray: Modified Reductive Perturbation Method as Applied to 7 ˆφ ˆφ = kv d ˆψ dξ =, at z = h, + k2 ˆφ ξ d ˆψ, on z = ˆψ(ξ) dξ kv ˆφ ξ + 2 [k2 ( ˆφ ξ )2 + ( ˆφ )2 ] + ˆψ =, on z = ψ(ξ). (52) We shall further assume that the wave number is small (lonwave) and the field variables ˆφ and ˆψ, and the wave speed V can be expanded into a power series in terms of the wave number k as: ˆφ = kφ + k 3 φ 2 + k 5 φ 3 + k 7 φ , ˆψ = k 2 ψ + k 4 ψ 2 + k 6 ψ 3 + k 8 ψ , V = V ( + k 2 V + k 4 V ) (53) Insertin the expansion (53) into the equations (5) and (52) and settin the coefficients of like powers of the wave number k equal to zero we obtain the followin set of differential equations and the boundary conditions: O(k) equations 2 φ =, (54) 2 φ =, at z =, φ = at z = h, O(k 3 ) equations ψ V φ ξ z= =. (55) 2 φ φ =, (56) ξ2 φ 2 =, at z = h, φ 2 z= + V dψ dξ =, O(k 5 ) equations ψ 2 V φ 2 ξ z= V V φ ξ z= + 2 ( φ ξ )2 z= =. (57) 2 φ φ 2 =, (58) ξ2 ( φ 3 + ψ 2 φ 2 2 ) z= + V ( dψ 2 dξ + V dψ dξ ) φ ξ dψ z= dξ =, ψ 3 [ V ( φ 3 ξ + ψ 2 φ 2 ξ ) + V φ 2 V ξ + V φ ] V 2 z= + ( φ ξ ξ O(k 7 ) equations φ 4 φ 2 ξ ) z= + 2 ( φ 2 )2 z= =. (59) 2 φ φ 3 =, (6) ξ2 =, at z = h, IJNS homepae:

8 8 International Journal of Nonlinear Science,Vol.6(28),No.,pp. -2 [ φ 4 +ψ 2 φ φ 2 2 ψ2 3 +ψ 2 φ ] z=+v ( dψ 3 dξ +V dψ 2 dξ +V dψ 2 dξ ) [ φ dψ 2 ξ dξ + φ 2 dψ ξ dξ ] z= =, ψ 4 [V ( φ 4 ξ +V 2 φ 3 ξ + 3 φ 2 2 ψ2 ξ 2 +ψ 2 φ 2 2 ξ ) V V ( φ 3 ξ +ψ 2 φ 2 ξ ) V φ 2 V 2 ξ V φ V 3 ξ ] z= +[ 2 ( φ 2 ξ )2 + φ ξ ( φ 3 ξ + ψ 2 φ 2 ξ )] z= + φ 2 ( φ 3 + ψ 2 φ 2 2 ) z= =. (6) 3. Solution of the field equations From the solution of the set iven in equations (54) and (55) we obtain φ = F (ξ), ψ = V where F (ξ) is an arbitrary function of its arument and to be determined later. Similarly, from the solution of the equations (56) and (57) one has df dξ, (62) φ 2 = d 2 F 2 dξ 2 (z2 + 2hz) + G(ξ), V = (h) /2, ψ 2 = V dg dξ + V V df dξ 2 (df dξ )2, (63) where G(ξ) is another unknown function whose overnin equation will be obtained later. From the solution of equations (58) and (59) (O(k 5 ) order equations) we have ψ 3 = V φ 3 = d 4 F 24 dξ 4 (z4 + 4hz 3 8h 3 z) d 2 G 2 dξ 2 (z2 + 2hz) + H(ξ), dh dξ hψ dψ dξ + V ψ 2 + 3V 2h ψ2 + (V 2 V 2 )ψ h ψ ψ 2 2h 2 ψ3 h 2 (dψ dξ )2. (64) The use of the boundary condition (58) 2 yields d 4 F dξ 4 + 9V df d 2 F h 3 dξ dξ 2 6V d 2 F h 2 =, (65) dξ2 or, usin the equation (62) we obtain the overnin equation for ψ as d 3 ψ dξ h 3 ψ dψ dξ 6V dψ h 2 dξ =. (66) In the present work, we shall be concerned with the localized travellin waves, i.e., the function ψ and its various order derivatives vanish as ξ ±. Hence, interatin (66) with respect to ξ and usin the localization condition we obtain d 2 ψ dξ h 3 ψ2 6V h 2 ψ =. (67) We shall seek a solution to equation (67) of the followin form ψ = a sech 2 η, η = αξ (68) where a is the wave amplitude and α is a constant to be determined later. Introducin (68) into equation (67) and settin the coefficients of like powers of sechη equal to zero, we obtain α = ( 3a 4h 3 )/2, From the solution of equations (6) and (6) one has V = a 2h. (69) φ 4 = d 6 F 72 dξ 6 (z6 +6hz 5 4h 3 z 3 +96h 5 z)+ d 4 G 24 dξ 4 (z4 +4hz 3 8h 3 z) d 2 H 2 dξ 2 (z2 +2hz)+I(ξ) (7) IJNS for contribution: editor@nonlinearscience.or.uk

9 Hilmi Demiray: Modified Reductive Perturbation Method as Applied to 9 where I(ξ) is another unknown function to be determined from the solution of the hiher order expansions. The use of the boundary condition (6) 2 yields where the function S(ψ ) is defined by S(ψ ) = 2h2 5 d 3 ψ 2 dξ 3 6V dψ 2 h 2 dξ + 9 d h 3 dξ (ψ ψ 2 ) = ds(ψ ), (7) dξ d 4 ψ dξ 4 5 2h (ψ ) 2 4 h ψ d 2 ψ dξ 2 + V d 2 ψ dξ 2 3 h 4 ψ3 + 5V 2h 3 ψ2 + 3 h 2 (2V 2 V 2 )ψ. (72) Interatin the equation (7) with respect to ξ and utilizin the localization condition we have, d 2 ψ 2 dξ 2 6V h 2 ψ h 3 (ψ ψ 2 ) = S(ψ ). (73) In order to use the solution proposed in equation (68) it would be convenient to express the differential equation (73) in terms of η. The result will be as follows: ψ 2 + ( 2ψ 4)ψ 2 = a S(ψ ), (74) where a prime denotes the differentiation with respect to η and S(ψ ) is defined by S(ψ ) = 3a h ψ(4) 5 2h (ψ ) 2 4 h ψ ψ 2 + a 2h ψ 4 ah ψ3 + 5 h ψ2 + 4h a (2V 2 a2 4h 2 )ψ. (75) Introducin the solution (68) into (74) and (75) the followin differential equation is obtained ψ 2 + (2sech 2 η 4)ψ 2 = 6a2 h sech6 η + 2a2 h sech4 η(8hv 2 9a2 5h )sech2 η (76) To obtain the travellin wave solution to this order of equation, we shall propose the function ψ 2 as ψ 2 = A sech 4 η + B sech 2 η (77) where A and B are two constants to be determined from the solution of (76). Carryin out the derivative of ψ 2 we have ψ 2 = 2Asech 6 η + (6A 6B)sech 4 η + 4Bsech 2 η. (78) Insertin (78) into equation (76) and settin the coefficients of sech 6 η and sech 4 η equal to zero, one obtains A = 3a2 4h, The remainin part of the equation (76) reads B = a2 2h. (79) (8hV 2 9a2 5h sech2 η = (8) If the constant V 2 was equal to zero, the equation would not be balanced and the solution will contain a secular term. In order to remove the secular term, the equation (8) must be satisfied identically. Thus, we obtain V 2 = 9a2 4h 2. (8) Hence, in terms of the real physical entities, the final solution up to this order may be iven as follows: with ˆψ = k 2 a sech 2 η + k 4 ( 3a2 4h sech4 η + a2 2h 2 sech2 η), (82) η = k( 3a 4h 3 )/2 [x V ( + a 2h k2 + 9a2 4h 2 k4 )t]. (83) This solution is exactly the same with the one iven in equations (48) and (49). As miht be seen from these formulations, the correction term to the soliton speed can be obtained quite easily. For instance, the O(ɛ 2 ) order correction term is 9a2, which corresponds to /A 4h 2 2 in Kraenkel et al[6] formulation, wherein A 2 remains to be undetermined. By use of this formulation the hiher order corrections term can be obtained without any principal difficulty. IJNS homepae:

10 2 International Journal of Nonlinear Science,Vol.6(28),No.,pp Concludin Remarks The study of the effects of hiher order terms in the perturbation expansion of field quantities throuh the use of classical reductive perturbation method leads to some secularities. To eliminate such secularities various methods, like renormalization method of Kodama and Taniuti [4], the multiple scale expansion method by Kraenkel et al[6], have been presented in the current literature. The results of present work and of those iven in references [8] and [9] proved that the modified reductive perturbation method presented by us is the most simplest and the effective one. By use of this method, any order of correction term may be obtained without any principal difficulties. References [] N. Davidson: Methods in nonlinear plasma theory. Academic Press. New York(972) [2] T. Taniuti: Reductive perturbation method and far field of wave equations. Proress in Theoretical Physics Supplement.55: -35(974) [3] N. Suimoto, T. Kakutani: Note on hiher order terms in reductive perturbation theory. J. Phys. Soc. Japan. 43: (977) [4] Y. Kodama, T. Taniuti: Hiher order approximation in reductive perturbation method, Weakly dispersive system. J. Phys. Soc. Japan. 45: 298-3(978) [5] H. Washimi, T. Taniuti: Propaation of ionic-acoustic solitary waves of small amplitude. Phys. Rev. Letter. 7: (966) [6] R. A. Kraenkel: M. A. Manna, J. G. Pereira: The Kortewe-deVries hierarchy and lon water waves. J. Math. Phys.. 36: 37-32(995) [7] M. Malfliet, E. Wieers: The theory of nonlinear ion-acoustic waves revisited. J. Plasma Phys.. 56: (996) [8] H. Demiray: A modified reductive perturbation method as applied to nonlinear ion-acoustic waves. J. Phys. Soc. Japan. 68: (999) [9] H. Demiray: On the contribution of hiher order terms to solitary waves in fluid-filled elastic tubes. Z. Anew. Math. Phys. (ZAMP). 5: 75-9(2) [] L. Wan, J. Zhou, L. Ren: The exact solitary wave solutions for a family of BBM equation. Int. J. Nonlinear Sci. : 58-64(26) [] D. Lu, B. Hon, L. Tian: Backlund transformation and N-soliton solution to the combined KdV- Burers equation with variable coefficien. Int. J. Nonlinear Sci.2:3-(26) [2] A. Bekir, E. Yusufolu: On the extended Tanh method applications of nonlinear equations, Int. J. Nonlinear Sci. 4: -6(27) [3] L. Wan: New exact solution for MNV equation. Int. J. Nonlinear Sci. 5: 52-57(28) IJNS for contribution: editor@nonlinearscience.or.uk