EXACT SOLUTIONS OF THE GENERALIZED POCHHAMMERCHREE EQUATION WITH SIXTHORDER DISPERSION


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1 EXACT SOLUTIONS OF THE GENERALIZED POCHHAMMERCHREE EQUATION WITH SIXTHORDER DISPERSION HOURIA TRIKI, ABDELKRIM BENLALLI, ABDULMAJID WAZWAZ 2 Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 2, Annaba, Algeria 2 Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA Received January 6, 205 Nonlinear waves described by the generalized Pochhammer Chree equation are analytically investigated. The addition of the sixthorder dispersion term, which will drastically change the characteristics of the equation, is examined. We employ the sinecosine method to derive a variety of exact solutions of distinct physical structures including solitons, compactons, periodic solitary pattern solutions for the adopted model, in the presence of the sixthorder dispersion term. The solitary wave ansatz method is used as well to obtain bright, singular, dark soliton solutions. Parametric conditions for the existence of the exact solutions are given. The obtained results show that the generalized Pochhammer Chree equation with sixth order dispersion reveals the richness of explicit soliton periodic solutions. It should be noted that the study of a new model admitting solitontype solutions is very important these solutions will be useful for future research work. Key words: PochhammerChree equation; dispersion; solitons. PACS: Lk, Jr, Yv.. INTRODUCTION Fully nonlinear evolution equations NLEEs have become more interesting more important in various physics problems like the analysis of patterns in liquid drops [, 2], weakly nonlinear ionacoustic waves dynamics in a plasma [3, 4] many others [5 3]. Interestingly, the dynamics of nonlinear waves described by these models have richer phenomena than the regular case, since various physical structures may be found as cidates for such propagating waves. In 993, Rosenau Hymann [] introduced the socalled Km,n equations to underst the role played by the nonlinear dispersion in the formation of patterns in liquid drops. This first class of fully NLEEs is a straightforward generalization of the well known Korteweg de Vries equation. Subsequently, many other NLEEs were generalized, offering a rich knowledge on the wave dynamics in nonlinear systems of all kinds. Examples of the many equations that have been generalized are the fully Boussinesq equation Bm,n [4, 5], the Zakharov Kuznetsov equation ZKm,n,k RJP Rom. 60Nos. Journ. Phys., 78, Vol , Nos. 78, 205 P , c 205 Bucharest,  v..3a*
2 936 Houria Triki, Abdelkrim Benlalli, AbdulMajid Wazwaz 2 [6], the modified Korteweg de Vriessine Gordon equation [7], the Km,n,p,q+r equation [8] so on. Investigation of exact solutions of fully NLEEs is important from many points of view e.g., for the calculation of certain physical quantities as well as serving as diagnostics for numerical simulations. Additionally, finding the explicit exact solutions for generalized NLEEs is particularly interesting in the study of nonlinear physical phenomena. Recently, many numerical analytical methods such as the sinecosine methods [9 2], the subsidiary ordinary differential equation method [22 24], Hirota s method [25], the Petrov Galerkin method [26], the collocation method [27], the solitary wave ansatz method [28, 29] many others, have been successfully applied to exactly solve NLEEs their generalizations. The generalized Pochhammer Chree GPC equation given by [30] u tt u ttxx αu + βu n+ + u 2n+ = 0, xx where α, β, are constants, represents a nonlinear model of longitudinal wave propagation of elastic rods [3 43]. Considering the higherorder dispersion terms in a given NLEE can significantly change its characteristics. In this work we extend the studies of the nonlinearly dispersive partial differential GPC equation to consider higher order dispersive effects. In particular, we focus our attention on the role of the sixth order dispersion term on the integrability of the GPC equation within the framework of the following model: u tt εu ttxx αu + κu n+ + u 2n+ xx δu xxxxxx = 0, 2 which can be considered as a natural extension of the generalized Pochhammer Chree equation. Here in Eq. 2, ε, α, κ,, δ are nonzero arbitrary constants, while the exponent n> is the power law nonlinearity parameter. In the case δ = 0, namely, in the absence of sixth order dispersion effect for ε =, Eq. 2 reduces to the usual GPC equation. To the best of our knowledge, the above sixth order GPC 6GPC equation 2 has not been previously considered in literature. The purpose of the present work is to establish fundamental analytical results regarding Eq. 2. Interestingly, many new families of exact travelling wave solutions of the adopted model are successfully obtained using the sinecosine the solitary wave ansatz methods. 2. EXACT SOLUTIONS In this section, we will use the sinecosine the solitary wave ansatz methods to develop exact travelling wave solutions to the 6GPC equation 2 with arbitrary constant coefficients. RJP 60Nos. 78, c v..3a*
3 3 Exact solutions of generalized PochhammerChree eq. with 6thorder dispersion 937 First, the wave variable ξ = x ct converts the NLEE 2 into an nonlinear ordinary differential equation ODE: c 2 u ξξ εc 2 u ξξξξ αu + κu n+ + u 2n+ ξξ δu ξξξξξξ = 0, 3 where u ξ denotes du/dξ. After integrating twice, Eq. 3 becomes c 2 α u εc 2 u ξξ κu n+ u 2n+ δu ξξξξ = 0, 4 where the integration constants are considered to be equal to zero. 2.. THE SINECOSINE METHOD The sinecosine ansatz admits the use of the following assumption for solving the reduced ODE Eq. 4 [20] or the assumption λcos β µξ }, ξ π 2µ, uξ = 0, otherwise λsin β µξ }, ξ π µ, uξ = 0, otherwise where λ, µ, β are parameters that will be determined, µ c are the wave number the wave speed, respectively. The exponent β will be determined as a function of n. The assumption 5 gives u n+ ξ = λ n+ cos βn+ µξ, u 2n+ ξ = λ 2n+ cos β2n+ µξ, 8 u ξξ = µ 2 β 2 λcos β µξ + µ 2 λβ β cos β2 µξ, 9 u ξξξξ =µ 4 β 4 λcos β µξ 2µ 4 λβ β β 2 2β + 2 cos β2 µξ + µ 4 λβ β β 2β 3cos β4 0 µξ, the derivatives of 6 become u n+ ξ = λ n+ sin βn+ µξ, RJP 60Nos. 78, c v..3a*
4 938 Houria Triki, Abdelkrim Benlalli, AbdulMajid Wazwaz 4 u 2n+ ξ = λ 2n+ sin β2n+ µξ, 2 u ξξ = µ 2 β 2 λsin β µξ + µ 2 λβ β sin β2 µξ, 3 u ξξξξ =µ 4 β 4 λsin β µξ 2µ 4 λβ β β 2 2β + 2 sin β2 µξ + µ 4 λβ β β 2β 3sin β4 µξ, Substituting Eqs. 70 into the reduced ODE 4 gives c 2 α λcos β µξ + εc 2 µ 2 β 2 λcos β µξ εc 2 µ 2 λβ β cos β2 µξ 4 κλ n+ cos βn+ µξ λ 2n+ cos β2n+ µξ δµ 4 β 4 λcos β µξ + 2δµ 4 λβ β β 2 2β + 2 cos β2 µξ δµ 4 λβ β β 2β 3cos β4 µξ = 0. Equating the exponents the coefficients of like powers of cosine function in Eq. 5 leads to 5 β β β 2β 3 0, 6 β 4 = β 2n +, 7 c 2 α λ + εc 2 µ 2 β 2 λ δµ 4 β 4 λ = 0, 8 εc 2 µ 2 λβ β κλ n+ + 2δµ 4 λβ β β 2 2β + 2 = 0, 9 Solving this system yields λ 2n+ δµ 4 λβ β β 2β 3 = 0, 20 β 0,,2,3, 2 µ = n 2 µ = n 2 δ n 2 + 2n + 2 β = 2 n, 22 εc 2 + ε 2 c 4 + 4δ c 2 α, 23 2δ εc 2 + κ RJP 60Nos. 78, c v..3a* δ n + 3n + 2 n + 2, 24
5 5 Exact solutions of generalized PochhammerChree eq. with 6thorder dispersion 939 where λ n,δ = εc 2 + κ λ = 2δ n 2 + 2n + 2 λ n,δ } n, 25 δ n + 3n + 2 δ n + 2n + 3n n + 2 Equating the two values of µ from Eqs gives nn + 2εc 2 +n 2 + 2n + 2 ε 2 c 4 + 4δc 2 α=2κ δn + 3n + 2, 27 n + 2 which serves as a constraint relation between the model coefficients the wave speed c provided that ε 2 c 4 + 4δ c 2 α > 0 δ < THE PERIODIC SOLUTIONS By inserting the previous results in 5 6, we obtain the following periodic solutions ux,t: ux,t = } n 2δ n 2 + 2n + 2 Γn,δ sec 2 n n 2 δ n 2 εc + 2n κ δ n + 3n + 2 n + 2 x ct, 28 } ux,t = 2δ n 2 + 2n + 2 Γn,δ n csc 2 n n 2 δ n 2 εc + 2n δ n + 3n κ x ct, 29 n + 2 where Γn,δ = εc 2 + κ δ n + 3n + 2 δ n + 2n + 3n + 2 n + 2 Note that the solutions exist provided that δ εc 2 + κ 30 δ n + 3n + 2 > 0 3 n + 2 RJP 60Nos. 78, c v..3a*
6 940 Houria Triki, Abdelkrim Benlalli, AbdulMajid Wazwaz 6 δ < 0. However, for δ εc 2 κ ux,t = 2.3. SOLITON SOLUTIONS δn+3n+2 n+2 2δ n 2 + 2n + 2 Γ n,δ sech 2 n n 2 δ n 2 εc + 2n κ < 0, the soliton solutions for ux,t } n δ n + 3n + 2 n + 2 x ct, 32 } ux,t = 2δ n 2 + 2n + 2 Γ n n,δ csch 2 n n 2 δ n 2 εc + 2n δ n + 3n κ x ct, 33 n + 2 are readily obtained, where Γ n,δ = εc 2 κ δ n + 3n + 2 n + 2 δn + 2n + 3n COMPACTON SOLUTIONS It is obvious that the periodic soliton solutions are obtained for n > 0. However, for negative exponent, we set n = m, where m is an integer, m > 0. In this case, Eqs become β 4 = β 2m +, 35 c 2 α λ + εc 2 µ 2 β 2 λ δµ 4 β 4 λ = 0, 36 εc 2 µ 2 λβ β κλ m+ + 2δµ 4 λβ β β 2 2β + 2 = 0, 37 λ 2m+ δµ 4 λβ β β 2β 3 = 0, 38 RJP 60Nos. 78, c v..3a*
7 7 Exact solutions of generalized PochhammerChree eq. with 6thorder dispersion 94 Solving the above system we get µ = m 2 µ = m 2 δ m 2 2m + 2 β = 2 m, 39 εc 2 + ε 2 c 4 + 4δ c 2 α, 40 2δ εc 2 + κ δ m2 3m 2 m, 4 } λ = 2δ m 2 2m + 2 Γ m 2m,δ, 42 Now substituting into 5 6 we obtain a family of compacton solutions given by } u = 2δm 2 2m+2 Γ m 2m,δ cos 2 m [µx ct], µξ π 2, 43 0 otherwise, where Γ 2 m,δ= } u = 2δm 2 2m+2 Γ m 2m,δ sin 2 m [µx ct], µξ π, 0 otherwise, εc 2 +κ 44 δ m2 3m δ2 m m2 3m m the wave speed c is obtained from equating the two values of the wave number µ from 40 4 as εc 2 mm 2 + m 2 2m + 2 ε 2 c 4 + 4δ c 2 α = 2κ δ m2 3m m Note that the solutions exist provided that δ εc 2 δ m2 3m + κ > m δ < RJP 60Nos. 78, c v..3a*
8 942 Houria Triki, Abdelkrim Benlalli, AbdulMajid Wazwaz THE SOLITARY PATTERN SOLUTIONS For δ εc 2 κ δm23m 2m < 0, we obtain a family of solitary pattern solutions given by } ux,t = 2δ m 2 2m + 2 µ m 3δ,m cosh 2 m m 2 δm 2 εc 2m δ m2 3m κ x ct, 49 2 m } ux,t = 2δ m 2 2m + 2 µ m 4δ,m sinh 2 m m 2 δm 2 εc 2m δ m2 3m + κ x ct, 50 2 m where µ 3 δ,m= µ 4 δ,m= εc 2 κ εc 2 + κ δ m2 3m δ2 m m2 3m, 5 2 m δ m2 3m δ2 m m2 3m, 52 2 m where m > 0 the wave speed c can be determined using Eq THE SOLITARY WAVE ANSATZ METHOD It is always useful desirable to construct exact solutions, especially solitontype envelope solutions for the understing of most nonlinear physical phenomena. In this section we will apply the solitary wave ansatz method to obtain the bright, singular, dark soliton solutions of the GPC equation in presence of the sixthorder dispersion term. RJP 60Nos. 78, c v..3a*
9 9 Exact solutions of generalized PochhammerChree eq. with 6thorder dispersion BRIGHT SOLUTIONS To obtain the bright soliton solution of Eq. 4, we adopt a soliton ansatz of the form [28, 29] A uξ = cosh p 53 µξ where ξ = x ct p > 0 for solitons to exist. Here, A is the amplitude of the soliton, c is the wave velocity µ is the inverse width of the soliton. The exponent p is unknown at this point its value will fall out in the process of deriving the solution of this equation. Thus from 53, we get u n+ ξ = u 2n+ ξ = A n+ cosh pn+ µξ, 54 A 2n+ cosh p2n+ µξ, 55 u ξξ = p2 Aµ 2 pp + Aµ2 cosh p µξ cosh p+2 µξ, 56 u ξξξξ = Ap4 µ 4 Aµ4 pp + p 2 + p + 2 2} cosh p µξ cosh p+2 µξ + Aµ4 pp + p + 2p + 3 cosh p+4, 57 µξ Substituting Eqs into Eq. 4 yields c 2 α A cosh p µξ εc2 p 2 Aµ 2 cosh p µξ + εc2 pp + Aµ 2 cosh p+2 µξ κa n+ A 2n+ cosh pn+ µξ cosh p2n+ µξ δap4 µ 4 cosh p µξ δaµ 4 pp + p 2 + p + 2 2} + cosh p+2 δaµ4 pp + p + 2p + 3 µξ cosh p+4 = 0, 58 µξ Now, from Eq. 58, equating the exponents p2n + p + 4 leads to which gives p2n + = p p = 2 n RJP 60Nos. 78, c v..3a*
10 944 Houria Triki, Abdelkrim Benlalli, AbdulMajid Wazwaz 0 It is clear that the same value of p can be obtained if the exponents pn + p + 2 are equated. Therefore, from Eq. 58, the linearly independent functions are /cosh p+j τ, where j = 0,2,4. Now setting the coefficients of the linearly independent functions to zero, we get the following expressions: µ = n εc 2 + ε 2 c 4 + 4δ c 2 α 6 2 2δ µ = n 2 δ n 2 εc + 2n δ n + 3n + 2 κ 62 n + 2 A n = 2δ n 2 εc 2 δ n + 3n + 2 κ + 2n + 2 n + 2 δn + 2n + 3n + 2 Therefore we obtain from Eqs the following expression: nn + 2εc 2 + n 2 + 2n + 2 ε 2 c 4 + 4δ c 2 α = 2κ 63 δ n + 3n n + 2 Having obtained the expressions for the pulse parameters, we construct a family of bright solitary wave solutions for the wave equation 2 as follows } ux,t = 2δ n 2 + 2n + 2 µ n 2 5δ,n sech n [iµx ct], 65 where µ 5 δ,n = εc 2 δ n + 3n + 2 δn + 2n + 3n + 2 κ, 66 n + 2 which exist provided that δ < 0 δ εc 2 κ < 0. δn+3n+2 n SINGULAR SOLUTIONS To find the singular soliton solution for the reduced ODE 4, we adopt an ansatz solution in the form uξ = Acsch p µξ, 67 RJP 60Nos. 78, c v..3a*
11 Exact solutions of generalized PochhammerChree eq. with 6thorder dispersion 945 where A µ are the amplitude the inverse width of the soliton, respectively, while ξ = x ct with c is the velocity of the wave. The unknown exponent p > 0 will be determined in term of the nonlinearity exponent n. From Eq. 67, it is possible to obtain u n+ ξ = A n+ csch pn+ µξ, 68 u 2n+ ξ = A 2n+ csch p2n+ µξ, 69 u ξξ = p 2 Aµ 2 csch p µξ + pp + Aµ 2 csch p+2 µξ, 70 u ξξξξ = Ap 4 µ 4 csch p µξ + Aµ 4 pp + + Aµ 4 pp + p + 2p + 3csch p+4 µξ Substituting Eqs into Eq. 4, gives c 2 α A csch p µξ εc 2 p 2 Aµ 2 csch p µξ εc 2 pp + Aµ 2 csch p+2 µξ p 2 + p + 2 2} csch p+2 µξ κa n+ csch pn+ µξ A 2n+ csch p2n+ µξ δap 4 µ 4 csch p µξ δaµ 4 pp + p 2 + p + 2 2} csch p+2 µξ δaµ 4 pp + p + 2p + 3csch p+4 µξ = 0, Now, from Eq. 72, equating the exponents p2n + p + 4 leads to so that 7 72 p2n + = p p = 2 74 n which is also obtained by equating the exponents pn + p + 2. Again from Eq. 72 the linearly independent functions are csch p+j τ for j = 0,2,4. Hence setting their respective coefficients to zero yields µ = n εc 2 + ε 2 c 4 + 4δ c 2 α δ µ = n 2 δ n 2 εc + 2n δ n + 3n κ n + 2 RJP 60Nos. 78, c v..3a*
12 946 Houria Triki, Abdelkrim Benlalli, AbdulMajid Wazwaz 2 } n A = 2δ n 2 εc 2 δ n + 3n κ αn, δ + 2n + 2 n + 2 Finally, inserting the above expressions into Eq. 67 leads to the exact singular soliton solution of Eq. 2: } n ux,t = 2δ n 2 εc 2 δ n + 3n κ αn, δ + 2n + 2 n + 2 csch 2 n n 2 δ n 2 εc + 2n δ n + 3n κ x ct, n + 2 where δ n + 2n + 3n + 2 αn,δ = which exist provided that δ < 0 δ εc 2 + κ δn+3n+2 n+2 77 < 0. We observe that the soliton solutions are the same as obtained above by the sinecosine method DARK SOLUTIONS The dark solitons that are also known as topological solitons are also supported by the NLEEs. In this case it will be seen that for the case of the GPC equation with sixthorder dispersion term the dark solitons will exist under certain restrictions. Let us begin the analysis by assuming an ansatz solution of the from [29] uξ = A tanh p µξ, 78 where A µ are free parameters, ξ = x ct with c is the velocity of the wave. Also, the unknown exponent p > 0 will be determined during the course of the derivation of the soliton solution to Eq. 2. From the ansatz 78, we get u n+ ξ = A n+ tanh pn+ µξ, 79 u 2n+ ξ = A 2n+ tanh p2n+ µξ, 80 u ξξ = paµ 2 [ p tanh p2 µξ 2ptanh p µξ + p + tanh p+2 µξ ], 8 RJP 60Nos. 78, c v..3a*
13 3 Exact solutions of generalized PochhammerChree eq. with 6thorder dispersion 947 u ξξξξ = paµ 4 p p 2p 3tanh p4 µξ + paµ 4 p + p + 2p + 3tanh p+4 µξ 2pAµ 4 p 2 + p 2 2} p }tanh p2 µξ 2pAµ 4 p 2 + p + 2 2} p + }tanh p+2 µξ + paµ 4 4p 3 + p 2 p 2 + p + 2 p + 2 } tanh p µξ. Substituting the previous results into Eq. 4, gives c 2 α A tanh p µξ εc 2 paµ 2 [ p tanh p2 µξ 2ptanh p µξ + p + tanh p+2 µξ ] κa n+ tanh pn+ µξ A 2n+ tanh p2n+ µξ δpaµ 4 p p 2p 3tanh p4 µξ δpaµ 4 p + p + 2p + 3tanh p+4 µξ + 2δpAµ 4 p 2 + p 2 2} p }tanh p2 µξ + 2δpAµ 4 p 2 + p + 2 2} p + }tanh p+2 µξ δpaµ 4 4p 3 + p 2 p 2 + p + 2 p + 2 } tanh p µξ = 0. From Eq. 83, equating the exponents p2n + p + 4 gives so that p2n + = p p = 2 85 n Again this same value of p is obtained on equating the exponents pn + p + 2. Now from Eq. 83 the linearly independent functions are tanh p+j τ for j = 0,±2,±4. Hence setting their respective coefficients to zero yields A [ c 2 α + 2p 2 εc 2 µ 2 δpµ 4 4p 3 + p 2 p 2 + p + 2 p + 2 }] = 0 86 Ap [ εc 2 pµ 2 + 2δpµ 4 p 2 + p 2 2}] = 0 87 εc 2 paµ 2 p + κa n+ + 2δpAµ 4 p 2 + p + 2 2} p + } = 0 88 A 2n+ δpaµ 4 p + p + 2p + 3 = 0 89 δpaµ 4 p p 2p 3 = 0 90 RJP 60Nos. 78, c v..3a*
14 948 Houria Triki, Abdelkrim Benlalli, AbdulMajid Wazwaz 4 Solving both of Eqs yields Thus, from Eqs we obtain p = 9 n = 2 92 This shows that topological soliton solutions of the GPC equation with sixthorder dispersion term will exist for n = 2. There is no other value of the power law nonlinearity parameter n for which the topological solitons will exist. This is a very important observation. In this case, equation 2 becomes u tt εu ttxx αu + κu 3 + u 5 xx δu xxxxxx = 0, 93 Substituting p = into gives µ = εc 2 + ε 2 c 4 + 6δ c 2 α 4 δ µ = εc 20δ 2 + κ 6δ [ A = εc 2 + κ 6δ 6δ 0δ Equating the two values of µ from Eqs gives the condition: εc ε 2 c 4 + 6δ c 2 α = 4κ 6δ ] The latter serves as a constraint relation between the coefficients the soliton velocity. Substituting the previous results in Eq. 78 leads to the following topological soliton solution to the 6GPC equation 2: u= [ 0δ εc 2 + κ 6δ 6δ ] 2 tanh εc 20δ 2 + κ which exist provided that δ < 0 δ εc 2 + κ 6δ > 0. RJP 60Nos. 78, c v..3a* δ x ct 98
15 5 Exact solutions of generalized PochhammerChree eq. with 6thorder dispersion CONCLUSION We have considered a generalized Pochhammer Chree equation with sixthorder dispersion term, which is an extension of the regular generalized Pochhammer Chree equation. By means of sinecosine method, we have obtained a variety of distinct physical structures such as periodic solutions, soliton solutions, compacton solutions, solitary pattern solutions. The solitary wave ansatz method is also used to obtain bright, singular, dark soliton solutions. Also, the parametric conditions for the existence of the exact solutions are given. These solutions may be of significant importance for the explanation of some special physical phenomena arising in dynamical systems modeled by the generalized Pochhammer Chree equation with higherorder dispersion effects. This study highlights the power of these methods for the determination of exact solutions to fully nonlinear evolution equations with arbitrary constant coefficients. REFERENCES. P. Rosenau J.M. Hyman, Compactons: Solitons with finite wavelength, Phys. Rev. Lett. 70 5, P. Rosenau D. Levy, Compactons in a class of nonlinearly quintic equations, Phys. Lett. A 252, S. Monro E.J. Parkes, The derivation of a modified Zakharov Kuznetsov equation the stability of its solutions, J. Plasma Phys. 62 3, H. Ma, Y. Yu, D. Ge, The auxiliary equation method for solving the Zakharov Kuznetsov equation, Computers Mathematics with Applications 58, H. Triki, Z. Jovanoski, A. Biswas, Dynamics of twolayered shallow water waves with coupled KdV equations, Rom. Rep. Phys. 66, Zhengping Yang, WeiPing Zhong, Analytical solutions to Sine Gordon equation with variable coefficient, Rom. Rep. Phys. 66, Lina Zhang Aiyong Chen, Exact loop solitons, cuspons, compactons smooth solitons for the Boussinesqlike B2,2 equation, Proc. Romanian Acad. A 5, R. Cimpoiasu, Symmetry reductions new wave solutions for the 2D Burgers Korteweg de Vries equation, Rom. J. Phys. 59, A. Biswas et al., Conservation laws of coupled Klein Gordon equations with cubic power law nonlinearities, Proc. Romanian Acad. A 5, P. Razborova, L. Moraru, A. Biswas, Perturbation of dispersive shallow water waves with RosenauKdVRLW equation power law nonlinearity, Rom. J. Phys. 59, A. Jafarian et al., Analytical approximate solutions of the Zakharov Kuznetsov equations, Rom. Rep. Phys. 66, H. Triki, Solitons periodic solutions to the dissipationmodified KdV equation with timedependent coefficients, Rom. J. Phys. 59, A.M. Wazwaz A. Ebaid, A Study on couplings of the fifthorder integrable Sawada Kotera Lax equations, Rom. J. Phys. 59, Z. Yan, New families of solitons with compact support for Boussinesqlike Bm,n equations with RJP 60Nos. 78, c v..3a*
16 950 Houria Triki, Abdelkrim Benlalli, AbdulMajid Wazwaz 6 fully nonlinear dispersion, Chaos Solitons & Fractals 4 8, Y. Zhu, Exact special solutions with solitary patterns for Boussinesqlike Bm,n equations with fully nonlinear dispersion, Chaos, Solitons & Fractals 22, H. Leblond D. Mihalache, Models of few optical cycle solitons beyond the slowly varying envelope approximation, Phys. Rep. 523, H. Leblond D. Mihalache, Fewopticalcycle solitons: Modified Kortewegde Vries sine Gordon equation versus other nonslowlyvaryingenvelopeapproximation models, Phys. Rev. A 79, H. Triki A.M. Wazwaz, Solitary wave solutions for a Km,n,p,q+r equation with generalized evolution, International Journal of Nonlinear Science, 4, A.M. Wazwaz, Integrable couplings of the Burgers equation the Sharma Tasso Olver equation: Multiple kink solutions, Rom. Rep. Phys. 65, A.M. Wazwaz, Multiple kink solutions for the 2+ dimensional integrable Gardner equation, Proc. Romanian Academy A 5, Y. Yang, Z. Tao, F.R. Austin, Solutions of the generalized KdV equation with timedependent damping dispersion, Appl. Math. Comput. 26, X. Li M. Wang, A subode method for finding exact solutions of a generalized KdVmKdV equation with highorder nonlinear terms, Phys. Lett. A 36, H. Triki, A.M. Wazwaz, SubODE method soliton solutions for the variablecoefficient mkdv equation, Appl. Math. Comput. 24, H. Triki T. Taha, The subode method soliton solutions for a higher order dispersive cubicquintic nonlinear Schrödinger equation, Chaos, Solitons & Fractals 42, K. Nakkeeran, Exact dark soliton solutions for a family of N coupled nonlinear Schrödinger equations in optical fiber media, Phys. Rev. E 64, M.S. Ismail, Numerical solution of complex modified Kortewegde Vries equation by Petrov Galerkin method, Appl. Math. Comput. 202, M. Dehghan A. Shokri, A numerical method for solution of the twodimensional sinegordon equation using the radial basis functions, Comput. Math. Simulation 79, A. Biswas, soliton solution of the Km,n equation with generalized evolution, Phys. Lett. A 372, H. Triki A.M. Wazwaz, Bright dark soliton solutions for a Km,n equation with t dependent coefficients, Phys. Lett. A 373, K. Par J. Rad, Some solitary wave solutions of generalized Pochhammer Chree equation via Expfunction method, World Academy of Science, Engineering Technology 43, C. Chree, Longitudinal vibrations of a circular bar, Quarterly Journal of Mathematics 2, L. Pochhammer, Biegung des kreiscylindersfortpflanzungsgeschwindigkeit kleiner schwingungen in einem kreiscylinder, Journal für die Reine und Angewte Mathematik 8, W.L. Zhang, Solitary wave solutions kink wave solutions for a generalized PC equation, Acta Mathematicae Applicatae Sinica 2, N. Shawagfeh D. Kaya, Series solution to the Pochhammer Chree equation comparison with exact solutions, Computers Mathematics with Applications 47, B. Li, Y. Chen, H. Zhang, Travelling wave solutions for generalized Pochhammer Chree equations, Zeitschrift für Naturforschung  Section A 57, Y. Liu, Existence blow up of solutions of a nonlinear Pochhammer Chree equation, Indiana University Mathematics Journal 45, I.L. Bogolubsky, Some examples of inelastic soliton interaction, Computer Physics Communica RJP 60Nos. 78, c v..3a*
17 7 Exact solutions of generalized PochhammerChree eq. with 6thorder dispersion 95 tions 3, P.A. Clarkson, R.J. LeVaque, R. Saxton, Solitary wave interactions in elastic rods, Studies in Applied Mathematics 75, A. Parker, On exact solutions of the regularized longwave equation: a direct approach to partially integrable equations. I. Solitary wave solitons, J. Math. Phys. 36, W.G. Zhang W.X. Ma, Explicit solitarywave solutions to generalized Pochhammer Chree equations, Journal of Applied Mathematics Mechanics 20, L. Jibin Z. Lijun, Bifurcations of travelling wave solutions in generalized Pochhammer Chree equation, Chaos, Solitons & Fractals 4, Z. Feng, On explicit exact solutions for the Lienard equation its applications, Phys. Lett. A 293, A.M. Wazwaz, The tanhcoth the sinecosine methods for kinks, solitons, periodic solutions for the Pochhammer Chree equations, Applied Mathematics Computation 95, RJP 60Nos. 78, c v..3a*