EXACT SOLUTIONS OF THE GENERALIZED POCHHAMMER-CHREE EQUATION WITH SIXTH-ORDER DISPERSION

EXACT SOLUTIONS OF THE GENERALIZED POCHHAMMER-CHREE EQUATION WITH SIXTH-ORDER DISPERSION HOURIA TRIKI, ABDELKRIM BENLALLI, ABDUL-MAJID WAZWAZ 2 Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 2, 23000 Annaba, Algeria E-mail: houriatriki@gmail.com, a-benlalli@yahoo.fr 2 Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA E-mail: wazwaz@sxu.edu Received January 6, 205 Nonlinear waves described by the generalized Pochhammer Chree equation are analytically investigated. The addition of the sixth-order dispersion term, which will drastically change the characteristics of the equation, is examined. We employ the sine-cosine 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 sixth-order 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 soliton-type solutions is very important these solutions will be useful for future research work. Key words: Pochhammer-Chree equation; dispersion; solitons. PACS: 02.30.Lk, 02.30.Jr, 05.45.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 ion-acoustic 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 so-called 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., 7-8, Vol. 935 95 60, Nos. 7-8, 205 P. 935 95, c 205 Bucharest, - v..3a*205.9.2

936 Houria Triki, Abdelkrim Benlalli, Abdul-Majid Wazwaz 2 [6], the modified Korteweg de Vries-sine 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 sine-cosine 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 higher-order 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 sine-cosine the solitary wave ansatz methods. 2. EXACT SOLUTIONS In this section, we will use the sine-cosine the solitary wave ansatz methods to develop exact travelling wave solutions to the 6GPC equation 2 with arbitrary constant coefficients. RJP 60Nos. 7-8, 935 95 205 c 205 - v..3a*205.9.2

3 Exact solutions of generalized Pochhammer-Chree eq. with 6th-order 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 SINE-COSINE METHOD The sine-cosine 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, 7 5 6 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. 7-8, 935 95 205 c 205 - v..3a*205.9.2

938 Houria Triki, Abdelkrim Benlalli, Abdul-Majid 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. 7-0 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. 7-8, 935 95 205 c 205 - v..3a*205.9.2 δ n + 3n + 2 n + 2, 24

5 Exact solutions of generalized Pochhammer-Chree eq. with 6th-order 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 + 2. 26 n + 2 Equating the two values of µ from Eqs. 23 24 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 < 0. 2.2. THE PERIODIC SOLUTIONS By inserting the previous results 22-25 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 + 2 2 + 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 + 2 2 δ n + 3n + 2 + 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 28 29 exist provided that δ εc 2 + κ 30 δ n + 3n + 2 > 0 3 n + 2 RJP 60Nos. 7-8, 935 95 205 c 205 - v..3a*205.9.2

940 Houria Triki, Abdelkrim Benlalli, Abdul-Majid 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 + 2 2 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 + 2 2 δ n + 3n + 2 + 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 + 2. 34 2.4. 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. 7-20 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. 7-8, 935 95 205 c 205 - v..3a*205.9.2

7 Exact solutions of generalized Pochhammer-Chree eq. with 6th-order 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 35-38 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. 45 2 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. 46 2 m Note that the solutions 43 44 exist provided that δ εc 2 δ m2 3m + κ > 0 47 2 m δ < 0. 48 RJP 60Nos. 7-8, 935 95 205 c 205 - v..3a*205.9.2

942 Houria Triki, Abdelkrim Benlalli, Abdul-Majid Wazwaz 8 2.5. 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 + 2 2 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 + 2 2 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. 46. 3. 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. 7-8, 935 95 205 c 205 - v..3a*205.9.2

9 Exact solutions of generalized Pochhammer-Chree eq. with 6th-order dispersion 943 3.. 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. 54-57 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 + 4 59 p = 2 n RJP 60Nos. 7-8, 935 95 205 c 205 - v..3a*205.9.2 60

944 Houria Triki, Abdelkrim Benlalli, Abdul-Majid 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 + 2 2 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. 6 62 the following expression: nn + 2εc 2 + n 2 + 2n + 2 ε 2 c 4 + 4δ c 2 α = 2κ 63 δ n + 3n + 2. 64 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+2 3.2. 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. 7-8, 935 95 205 c 205 - v..3a*205.9.2

Exact solutions of generalized Pochhammer-Chree eq. with 6th-order 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. 68-7 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 + 4 73 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 75 2 2δ µ = n 2 δ n 2 εc + 2n + 2 2 δ n + 3n + 2 + κ n + 2 RJP 60Nos. 7-8, 935 95 205 c 205 - v..3a*205.9.2 76

946 Houria Triki, Abdelkrim Benlalli, Abdul-Majid Wazwaz 2 } n A = 2δ n 2 εc 2 δ n + 3n + 2 + 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 + 2 + κ αn, δ + 2n + 2 n + 2 csch 2 n n 2 δ n 2 εc + 2n + 2 2 δ n + 3n + 2 + 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 56 68 are the same as obtained above by the sinecosine method. 3.3. 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 sixth-order 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. 7-8, 935 95 205 c 205 - v..3a*205.9.2

3 Exact solutions of generalized Pochhammer-Chree eq. with 6th-order 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 82 83 p2n + = p + 4 84 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. 7-8, 935 95 205 c 205 - v..3a*205.9.2

948 Houria Triki, Abdelkrim Benlalli, Abdul-Majid Wazwaz 4 Solving both of Eqs. 87 90 yields Thus, from Eqs. 80 9 we obtain p = 9 n = 2 92 This shows that topological -soliton solutions of the GPC equation with sixth-order 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 86-90 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. 94 95 gives the condition: εc 2 + 5 ε 2 c 4 + 6δ c 2 α = 4κ 6δ ] 2 94 95 96 97 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. 7-8, 935 95 205 c 205 - v..3a*205.9.2 6δ x ct 98

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