Badji Mokhtar University, P. O. Box 12, Annaba, Algeria. (Received 27 Februry 2011, accepted 30 May 2011)

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1 ISSN (print), (online) International Jornal of Nonlinear Science Vol.(20) No.4,pp Solitary Wave Soltions for a K(m,n,p,q+r) Eqation with Generalized Evoltion Horia Triki, Abdl-Majid Wazwaz 2 Radiation Physics Laboratory, Department of Physics, Faclty of Sciences, Badji Mokhtar University, P. O. Box 2, Annaba, Algeria. 2 Department of Mathematics, Saint Xavier University, Chicago, IL (Received 27 Febrry 20, accepted 30 May 20) Abstract: Stdying solitons compactons is of important significance in nonlinear physics. In this work we stdy an extension of the K(n,n) eqation the reslting compactons that appear in the sper deformed nclei, the fission of liqid drops the inertial fsion. The generalized eqation, with the generalized evoltion term, nonlinear convection terms, the fifth-order nonlinear dispersion the higher-order nonlinear dispersion corrections will be examined. The sine-cosine ansatz is sed to carry ot the analysis for this eqation to stdy the strctres of the obtained soltions. A variety of soltions of different strctres that contain periodic soltions, solitons soltions, compactons soltions solitary pattern soltions is obtained. Keywords: K(n,n) eqation; sine-cosine ansatz; solitary waves soltions; compactons AMS codes: 35Q5; 35Q53; 37K0. Introdction The celebrated KdV eqation reads t + 6 x + xxx = 0. () The KdV eqation models a variety of nonlinear phenomena, inclding ion acostic waves in plasmas, shallow water waves. The derivative t characterizes the time evoltion of the wave propagating in one direction, the nonlinear term x describes the steepening of the wave, the linear term xxx acconts for the spreading or dispersion of the wave. The KdV eqation () gives rise to solitons de to the balance between the nonlinear convection x the linear dispersion xxx. Soliton is a localized wave that has an infinite spport or a localized wave with exponential wings [ 5]. Solitons retain their identities after mtal collision. This means that soliton has the property of a particle. Moreover, the stard K(n, n) eqation t + ( n ) x + ( n ) xxx = 0, (2) where compactons arise as a reslt of the delicate balance between the nonlinear convection ( n ) x with genine nonlinear dispersion of ( n ) xxx. Compactons are solitary waves with exact compact spport that are termed compactons. Unlike soliton that narrows as the amplitde increases, the compacton s width is independent of the amplitde. In modern physics, a sffix-on is sed to indicate the particle property, for example phonon, photon, soliton. For this reason, the solitary wave with compact spport is called compacton to indicate that it has the property of a particle. The classical solitons are analytic soltions, whereas compactons are nonanalytic soltions. Compactons were proved to collide elastically [6 22] vanish otside a finite core region. However, the compacton concept has been stdied by sing many analytical nmerical methods, sch as the psedo spectral method, the tri-hamiltonian operators, the finite difference method, Adomian decomposition method. The compactons discovery motivated a considerable work to make compactons be practically realized in scientific applications, sch as the sper deformed nclei, pre-formation of clster in hydrodynamic models, the fission of liqid drops (nclear physics), inertial fsion others. Corresponding athor. address: halimgamil@yahoo.com Copyright c World Academic Press, World Academic Union IJNS /488

2 388 International Jornal of Nonlinear Science, Vol.(20), No.4, pp Compactons have mltiple applications in physics. The K(n,n) was discovered as a simplified model to stdy the role of nonlinear dispersion on pattern formation in liqid drops, being also proposed in the analysis of patterns on liqid srfaces. Eqations with compacton soltions have also fond applications sch as the lbrication approximation for thin viscos films [22], semi classical models for Bose Einstein condensates, long nonlinear srface waves in a rotating ocean when the high-freqency dispersion is nll, the plse propagation in ventricleaorta system, dispersive models for magma dynamics, or, even, particle wave fnctions in nonlinear qantm mechanics. In nonlinear lattices the propagation of compacton-like kinks has been observed sing mechanical, electrical, magnetic analogs. Nonlinear evoltion eqations spporting travelling wave soltions have been the sbject of intense research in recent years. The investigation of exact travelling wave soltions to non-linear partial differential eqations (NLPDEs) plays an important role in the stdy of non-linear physical phenomena []. Traveling waves appear in many physical strctres in solitary wave theory sch as solitons, kinks, peakons, cspons [2]. One of the localized travelling waves which has attracted a considerable interest in nonlinear science is envelope solitons becase of their potential application in many physics areas. Solitons are defined as localized waves that propagate withot change of its shape velocity properties stable against mtal collisions [3]. It needs to be noted that the formation of this kind of plses is de to a perfect balance between weak nonlinearity dispersion effects nder specific conditions. From a mathematical point of view, there exists a certain class of NLPDEs that spport soliton soltions in physical systems. The most famos ones are the nonlinear Schrödinger (NLS) eqation, the Korteweg-de Vries (KdV) eqation, the sine- Gordon (sg) eqation so on. In many practical physics problems, the reslting nonlinear wave eqations of interest are non integrable [4]. In some particlar cases they may be close to an integrable one [4]. It is remarkable that non-integrability is not necessarily related to the nonlinear terms [5]. Higher order dispersions, for example, also can make the system to be non-integrable (while it remains Hamiltonian) [5]. The effort in finding exact soltion to nonlinear eqation is important for the ndersting of most nonlinear physical phenomena [6]. In recent years, many powerfl methods to constrct exact soltions of NLPDEs have been established developed, which lead to one of the most excited advances of nonlinear science theoretical physics [7]. In fact, many kinds of exact soliton soltions have been obtained by sing for example, the homogeneos balance principle F- expansion method [8], the Jacobi elliptic fnctions method [6], the sine-cosine tanh methods [9], the Hirota s bilinear method, the Bäcklnd transformation method, so on. The concept of compactons: solitons with compact spport, or strict localization of solitary waves appeared recently in the literatre [0]. In [0], the athors introdced a geninely nonlinear dispersive eqation K(m, n), a special type of the KdV eqation, to examine the role played by the nonlinear dispersion in the formation of patterns in liqid drops. The proposed K(m, n) eqation takes the form [0] t + ( m ) x + ( n ) xxx = 0, m > 0, < n 3. (3) which is a generalization of the KdV eqation. As important examples, we find that the K(2, ) K(3, ) models are the well known KdV mkdv eqations. The delicate interaction between nonlinear convection ( m ) x with genine nonlinear dispersion ( n ) xxx in the K(m, n) eqation () generates solitary waves with exact compact spport that are termed compactons. The isse of the existence of compactons soltions for the K(m, n) models has been addressed by many athors since the analysis made so far in this regard. Wazwaz [] discssed two generalized forms of the K(n, n) the KP eqations that exhibit compactons. In [2], a stdy was condcted on mk(n, n) eqations in higher dimensional spaces the constrction of compact noncompact soltions was shown. Lately, Wazwaz [3] sed the Adomian decomposition method to constrct exact special soltions with solitary patterns to the defocsing K(m, n) eqation: t a ( m ) x + ( n ) xxx = 0, m, n >. (4) The new solitary wave special soltions with compact spport were also constrcted in [0] [4] for the nonlinear focsing dispersive K(m, n) eqation: t + a ( m ) x + ( n ) xxx = 0, m, n >. (5) Recently, Biswas [5] obtained the - soliton soltion of K(m, n) eqation with the generalized evoltion term of the form ( l ) t + am x + b ( n ) xxx = 0, (6) where a, b R are constants, while l, m, n Z +. IJNS for contribtion: editor@nonlinearscience.org.k

3 H. Triki, A. Wazwaz: Solitary Wave Soltions for a K(m,n,p,q+r) Eqation with Generalized Evoltion 389 More recently, Yin Tian [6] have sed the variational iteration method for solving the K(2, 2, ) K(3, 3, ) eqations that are particlar forms of the following nonlinear dispersive K(p, q, ) eqations In [7], the following two eqations: t + a ( p ) x + ( q ) 3x + 5x = 0, (7) t + ( m ) x + ( m ) 3x + δ ( m ) 5x = 0, (8) t + ( m+) x + [ (n ) xx ] x = 0. (9) were examined a variety of explicit compact solitary waves is determined. Very recently, in [8], the athors have stdied the two three dimensional, N = 2, 3, nonlinear dispersive eqation C N (m, a + b): t + ( m ) x + [ a ( 2 b)] = 0, (0) b x where the degeneration of the dispersion at the grond state indces cylindrically spherically symmetric compactons convected in the x direction. 2 The problem In this work, we introdce the following K(m, n, p, q + r) eqation with the generalized evoltion term, nonlinear convection terms, the fifth-order nonlinear dispersion higher-order nonlinear dispersions corrections: ( l ) t + a m x + a 2 m+ x + a 3 ( n ) 3x + a 4 ( p ) 5x + a 5 [ q ( r ) xx ] x = 0. () where a i R are constants, i =,..., 5, with l, m, n, p, q, r Z +. Note that the coefficients a i depend on the physical context. The finding of these coefficients is directly related to the environmental properties of the medim. Here in (), the first term is the generalized evoltion term, the second third terms together represent nonlinear advection, the forth fifth terms represent the third-order fifth-order nonlinear dispersions respectively, while the last term is a nonlinear dispersion correction term. Remarkably, the last term proportional to a 5 contains two nonlinear dispersion contribtions q ( r ) xxx ( q ) x ( r ) xx. This eqation is a generalized qintic extension of the K(m, n) eqation (4). In fact, when extremely short plses are considered, the stard K(m, n) eqation (4) fails in the physical description of the propagation of solitary waves higher-order effects mst be taken into accont. Ths, the dynamics of plses is described by the K(m, n) family of eqations with higher order nonlinear dispersions terms. Attention has been focsed on the role of additional terms (the third, fifth last terms) which describe higher order nonlinear dispersive effects may have inflence on the properties of the reslting soltions. As a matter of fact, this problem is important since the generalized versions of the K(m, n) eqation are of a great interest of both mathematical physical point of view. In the particlar case where a 2 = a 4 = a 5 = 0, Eq. () redces to the K(m, n) eqation (4) which has been interestingly stdied by Biswas [5]. In the limit l = m = n = p = q = r =, Eq. () becomes t + a x + a 2 2 x + a 3 3x + a 4 5x + a 5 [ () xx ] x = 0. (2) Formally, Eq. (0) is identical to the celebrated higher-order KdV eqation that has been extensively stdied by Marchant [9-20]. First, we shold point ot that Eq. () is not integrable becase of the presence of the higher-order nonlinear dispersive effects. By integrable [23 33] we mean that the eqation has Lax pairs gives mltiple soliton soltions. Also, integrable eqations have infinite laws of conservation of energy.the KdV eqation, modified KdV eqation, Schrodinger eqation, Benjamin-Ono eqation are examples of integrable eqations. The fifth-order Lax eqation, the fifth-order Sawada-Kotera eqation, the fifth-order Kap-Kperschnidt eqation, the Bossinesq eqation, the sixth-order Ramani eqations are also examples of integrable eqations [23 33]. Moreover, the Brgers eqation t + a x + ν xx, (3) that incldes nonlinear evoltion term the dissipative term xx also is an integrable eqation. IJNS homepage:

4 390 International Jornal of Nonlinear Science, Vol.(20), No.4, pp As stated before, the K(m,n) eqation is not an integrable eqation. The BBM eqation, the generalized KdV eqation, the Zakharov-Kznetsov eqation the fifth-order Kawahara eqation are examples of non-integrable eqations. Mltiple soliton soltions for these eqations cannot be obtained in this case. It is always sefl desirable to constrct exact analytical soltions by sing appropriate techniqes. In this paper, we deal with the existence of exact solitary wave soltions of the K(m, n, p, q + r) eqation (9) as it appears, namely for general vales of l, m, n, p, q r. Importantly, it is not possible to integrate (9) by the inverse scattering transform for any general vales of the exponents l, m, n, p, q r since the Painlevé test of integrability will fail in this sitation. It is widely believed that possession of the Painlevé property is a sfficient criterion for integrability [5]. In this work, we plan to se the sine-cosine ansatz for the determination of exact analytical solitary wave soltions for the considered nonlinear evoltion eqation. All the physical parameters in the solitary wave soltions are obtained as fnctions of the model coefficients. To the best of or knowledge, stdies of exact soltions for the K(m, n) models inclding higher-order effects are still few. Moreover, the K(m, n, p, q + r) model (9) with varios important physical effects general vales of the exponents l, m, n, p, q r was not introdced stdied before. It is worth noting that the existence or the non-existence of solitary wave soltions depends on the dependent model coefficients, therefore on the specific nonlinear dispersive featres of the medim. 3 Exact solitary wave soltions We begin or analysis by introdcing the wave variable: ξ = x ct, (4) where c is the wave speed. The transformation () converts the nonlinear eqation (9) into c ( l) + a ( m+ ) + a22 ( m+2 ) + a3 ( n ) + a 4 ( p ) + a 5 [ q ( r ) ] = 0, (5) where a = a m+ a 22 = a 2 m+2. Here the prime represents the derivative of (ξ) with respect to the variable ξ. Integrating Eq. (2) once give rise to the following redced ODE eqation c l + a m+ + a 22 m+2 + a 3 ( n ) + a 4 ( p ) + a 5 q ( r ) = 0, (6) Note that we have set the integration constant to zero for the case of solitary wave soltions. Compactons are compact soltions that are sally expressed by powers of trigonometric fnctions sine cosine [2]. The sine cosine ansatz admits the se of the assmption (ξ) = λ cos β (μξ) }, ξ π 2μ (7) or the assmption (ξ) = λ sin β (μξ) }, ξ π μ (8) where λ, μ β are parameters that will be determined. Here λ represents the amplitde of the compacton, while μ is the wave nmber. The exponent β will be determined as a fnction of l, m, n, p, q r. From the ansatz (5), one obtains l (ξ) = λ l cos βl (μξ), (9) m+ (ξ) = λ m+ cos β(m+) (μξ), (20) m+2 (ξ) = λ m+2 cos β(m+2) (μξ), (2) ( n ) = n 2 μ 2 β 2 λ n cos nβ (μξ) + nμ 2 λ n β (nβ ) cos nβ 2 (μξ), (22) ( p ) = μ 4 λ p β 4 p 4 cos βp (μξ) 2μ 4 λ p βp (βp ) ( β 2 p 2 2βp + 2 ) cos βp 2 (μξ) μ 4 λ p βp (βp ) (βp 2) (βp 3) cos βp 4 (μξ), (23) q ( r ) = μ 2 λ q+r βr (βr ) cos β(q+r) 2 (μξ) μ 2 λ q+r β 2 r 2 cos β(q+r) (μξ).. (24) IJNS for contribtion: editor@nonlinearscience.org.k

5 H. Triki, A. Wazwaz: Solitary Wave Soltions for a K(m,n,p,q+r) Eqation with Generalized Evoltion 39 From the ansatz (6), we get l = λ l sin βl (μξ), m+ = λ m+ sin β(m+) (μξ), (26) m+2 = λ m+2 sin β(m+2) (μξ), (27) ( n ) = n 2 μ 2 β 2 λ n sin nβ (μξ) + nμ 2 λ n β (nβ ) sin nβ 2 (μξ), (28) ( p ) = ( p ) = μ 4 λ p β 4 p 4 sin βp (μξ) 2μ 4 λ p βp (βp ) ( β 2 p 2 2βp + 2 ) sin βp 2 (μξ), μ 4 λ p βp (βp ) (βp 2) (βp 3) sin βp 4 (μξ), (29) q ( r ) = μ 2 λ q+r βr (βr ) sin β(q+r) 2 (μξ) μ 2 λ q+r β 2 r 2 sin β(q+r) (μξ). (30) Sbstitting (7) (22) into the redced ODE (4) gives cλ l cos βl (μξ) + a λ m+ cos β(m+) (μξ) + a 22 λ m+2 cos β(m+2) (μξ) +a 3 n 2 μ 2 β 2 λ n cos nβ (μξ) + nμ 2 λ n β (βn ) cos nβ 2 (μξ) } +a 4 μ 4 λ p β 4 p 4 cos βp (μξ) 2μ 4 λ p βp (βp ) ( β 2 p 2 2βp + 2 ) cos βp 2 (μξ) μ 4 λ p βp (βp ) (βp 2) (βp 3) cos βp 4 (μξ) } } +a 5 μ 2 λ q+r βr (βr ) cos β(q+r) 2 (μξ) μ 2 λ q+r β 2 r 2 cos β(q+r) (μξ) = 0. (3) To obtain solitary wave soltions for the K(m, n, p, q + r) eqation (9), we need to impose some restrictions on the dependent exponents so that Eq. (29) satisfies the homogeneos balance principle. In fact, a jdicios choice of exponents l, m, n, p, q r leads to a specific closed form soltion that is physically meaningfl. This may be a complicated task since we are concerned by a K(m, n, p, q + r) model with several dependent exponents. In what follows we analyze the reslting Eq. (29) in the framework of two interesting cases depending on the exponents of the considered model. 3. Case I: l = n = p = m + = q + r In this case, Eq. (29) becomes (25) cλ l cos βl (μξ) + a λ l cos βl (μξ) + a 22 λ l+ cos β(l+) (μξ) +a 3 l 2 μ 2 β 2 λ l cos βl (μξ) + lμ 2 λ l β (βl ) cos lβ 2 (μξ) } +a 4 μ 4 λ l β 4 l 4 cos βl (μξ) 2μ 4 λ l βl (βl ) ( β 2 l 2 2βl + 2 ) cos βl 2 (μξ) μ 4 λ l βl (βl ) (βl 2) (βl 3) cos βl 4 (μξ) } +a 5 μ 2 λ l βr (βr ) cos βl 2 (μξ) μ 2 λ l β 2 r 2 cos βl (μξ) } = 0. (32) Eqating the exponents the coefficients of like powers of cosine fnctions leads to Solving this system yields βl (βl ) (βl 2) (βl 3) = 0, (33) βl 4 = β (l + ), (34) c + a a 3 l 2 μ 2 β 2 + a 4 μ 4 β 4 l 4 a 5 μ 2 β 2 r 2 = 0, (35) a 3 lμ 2 β (βl ) 2a 4 μ 4 βl (βl ) ( β 2 l 2 2βl + 2 ) + a 5 μ 2 βr (βr ) = 0, (36) a 22 λ l+ a 4 μ 4 λ l βl (βl ) (βl 2) (βl 3) = 0. (37) βl = 0,, 2, 3, (38) β = 4, (39) μ = } 2 a 4 (8l 2 a 3 + a 5 (40) + 4l + ) l (4l + ) c = a 6 ( l 2 a 3 + r 2 a 5 ) μ a 4 μ 4 l 4, (4) λ = 4a 4μ 4 l (4l + ) (4l + 2) (4l + 3) a 22. (42) IJNS homepage:

6 392 International Jornal of Nonlinear Science, Vol.(20), No.4, pp In Eqs. (39) (40), μ is given by Eq. (38). Clearly, two sorts of solitary wave soltions can be obtained in the wave nmber expression (38). When } a 4 (8l 2 a 3 + a 5 < 0, + 4l + ) l (4l + ) the solitary wave soltions take the forms of the sec csc fnctions giving rise to the following periodic soltions: [ } ] (x, t) = λ sec 4 2 a 4 (8l 2 a 3 + a 5 (x ct), (43) + 4l + ) l (4l + ) However, in the opposite case where [ 2 (x, t) = λ csc 4 2 a 4 (8l 2 + 4l + ) the soliton soltions are obtained in the form [ 3 (x, t) = λ sec h 4 2 } a 4 (8l 2 a 3 + a 5 > 0, + 4l + ) l (4l + ) [ 4 (x, t) = λ csc h 4 2 where λ c are determined from Eqs. (39) (40). 3.2 Case II: p = q + r In this case, Eq. (29) becomes a 4 (8l 2 + 4l + ) a 4 (4l 2 + 4l + ) } ] a 3 + a 5 (x ct), (44) l (4l + ) } ] a 3 + a 5 (x ct), (45) l (4l + ) } ] a 3 + a 5 (x ct), (46) l (4l + ) cλ l cos βl (μξ) + a λ m+ cos β(m+) (μξ) + a 22 λ m+2 cos β(m+2) (μξ) +a 3 n 2 μ 2 β 2 λ n cos nβ (μξ) + nμ 2 λ n β (βn ) cos nβ 2 (μξ) } +a 4 μ 4 λ p β 4 p 4 cos βp (μξ) 2μ 4 λ p βp (βp ) ( β 2 p 2 2βp + 2 ) cos βp 2 (μξ) μ 4 λ p βp (βp ) (βp 2) (βp 3) cos βp 4 (μξ) } +a 5 μ 2 λ p βr (βr ) cos βp 2 (μξ) μ 2 λ p β 2 r 2 cos βp (μξ) } = 0. (47) To determine β, we sally balance the highest power of cos β(m+2) (μξ) cos βp 4 (μξ) terms in Eq. (45) so that one obtains 4 β = p m 2, (48) for p = m + 2. (49) It is important to note that the compacton soltion (5) exists only when β > 0. This condition implies that p > m + 2 in Eq. (46). Now setting the coefficients of cos βp 2 (μξ) terms to zero in Eq. (45) yields 2a 4 μ 4 λ p βp (βp ) ( β 2 p 2 2βp + 2 ) + a 5 μ 2 λ p βr (βr ) = 0, (50) which gives a 5 r (βr ) μ = 2a 4 p (βp ) (β 2 p 2 2βp + 2). (5) IJNS for contribtion: editor@nonlinearscience.org.k

7 H. Triki, A. Wazwaz: Solitary Wave Soltions for a K(m,n,p,q+r) Eqation with Generalized Evoltion 393 From eqating the exponents of cos nβ 2 (μξ) cos βl (μξ) terms in Eq. (45), we get β = 2 n l, (52) for n = l. (53) In this case, since β is positive for compactons soltions it follows directly that n > l. Then, we find from setting their corresponding coefficients to zero that cλ l + a 3 nμ 2 λ n β (βn ) = 0, (54) which leads to for λ = 2ca4 p (βp ) ( β 2 p 2 2βp + 2 ) a 5 a 3 nrβ (βn ) (βr ) } n l, (55) c = 0, βn = 0, βr = 0, βp = 0, n = l. (56) Also, eqating the two vales of β from (46) (50) leads to an algebraic relationship between the dependent model exponents: 2n + m + 2 = p + 2l, (57) which serves as a condition for the compactons to exist for the model (9) Compactons soltions By sing Eqs. (49), (50) (53), we obtain a family of compactons soltions [ ] 2ca 4 p(βp )(β 2 p 2 2βp+2) } (x, t) = a 5 a 3 nrβ(βn )(βr ) cos 2 a 5 r(βr ) 2a 4 p(βp )(β 2 p 2 2βp+2) (x ct) n l, x ct π 0, otherwise. 2μ (58) 2ca 4 p(βp )(β 2 p 2 2βp+2) [ ] } (x, t) = a 5a 3nrβ(βn )(βr ) sin 2 a 5r(βr ) 2a 4p(βp )(β 2 p 2 2βp+2) (x ct) n l, x ct π μ 0, otherwise. (59) Solitary patterns soltions By sing Eqs. (49), (50), (53), we obtain a family of solitary pattern soltions (x, t) = 2ca4 p (βp ) ( β 2 p 2 2βp + 2 ) a 5 a 3 nrβ (βn ) (βr ) cosh 2 φ } n l, (60) where (x, t) = 2ca 4p (βp ) ( β 2 p 2 2βp + 2 ) a 5 a 3 nrβ (βn ) (βr ) sinh 2 φ a 5 r (βr ) φ = 2a 4 p (βp ) (β 2 p 2 (x ct). 2βp + 2) } n l, (6) IJNS homepage:

8 394 International Jornal of Nonlinear Science, Vol.(20), No.4, pp Conclsion Compactons have mltiple applications in physics many scientific applications. The K(n,n) was discovered as a simplified model to stdy the role of nonlinear dispersion on pattern formation in liqid drops [22], being also proposed in the analysis of patterns on liqid srfaces. Eqations with compacton soltions have also fond applications sch as the lbrication approximation for thin viscos films [22], semi classical models for Bose Einstein condensates, long nonlinear srface waves in a rotating ocean when the high-freqency dispersion is nll, the plse propagation in ventricleaorta system, dispersive models for magma dynamics, or, even, particle wave fnctions in nonlinear qantm mechanics [22]. In nonlinear lattices the propagation of compacton-like kinks has been observed sing mechanical, electrical, magnetic analogs [22]. We have considered a K(m, n, p, q+r) eqation with the generalized evoltion term, nonlinear convection terms, fifthorder nonlinear dispersion nonlinear dispersions corrections. The examined model is a generalized qintic extension of the K(m, n) eqation. By means of sine-cosine method, we have obtained a variety of physical soltions, inclding periodic soltions, solitons soltions, compactons soltions solitary patterns soltions nder two sets of parametric conditions. These soltions may be sefl to explain some physical phenomena in geninely nonlinear dynamical systems spporting higher order nonlinear dispersions. References [] G. X, Z. Li Exact travelling wave soltions of the Whitham-Broer-Kap Broer-Kap-Kpershmidt eqations.chaos, Solitons Fractals, 24 :(2005), [2] A.M. Wazwaz. New solitary wave soltions to the modified Kawahara eqation Phys. Lett. A, 360: (2007), [3] A.M. Wazwaz. New solitons kink soltions for the Gardner eqation. Commn. Nonlinear Sci. Nmer. Siml., 2: (2007), [4] S.L. Palacios. Two simple ansätze for obtaining exact soltions of high dispersive nonlinear Schrödinger eqations. Chaos, Solitons Fractals, 9 :(2004), [5] S.L. Palacios, J.M. Fernez-Diaz. Black optical solitons for media with parabolic nonlinearity law in the presence of forth order dispersion. Opt. Commn., 78: (2000), [6] E.M.E. Zayed, H.A. Zedan, K.A. Gepreel. On the solitary wave soltions for nonlinear Hirota-satsma copled KdV of eqations. Chaos, Solitons Fractals, 22 :(2004), [7] X. Li, Mingliang Wang. A sb-ode method for finding exact soltions of a generalized KdV-mKdV eqation with high-order nonlinear terms. Phys. Lett A., 36: (2007), 5-8. [8] L. P. X, J. L. Zhang. Exact soltions to two higher order nonlinear Schrödinger eqations. Chaos, Solitons Fractals, 3:(2007), [9] A.M. Wazwaz. Analytic stdy for fifth-order KdV- type eqations with arbitrary power nonlinearities. Commn Nonlinear Sci Nmer Siml., 2: (2007), [0] P. Rosena, J. M. Hyman. Compactons: Solitons with finite wavelength. Phys. Rev. Lett., 70 (5):(993), [] A.M. Wazwaz. Compactons dispersive strctres for variants of the K(n, n) the KP eqations. Chaos, Solitons Fractals, 3: (2002), [2] A.M. Wazwaz. General compactons soltions solitary patterns soltions for modified nonlinear dispersive eqations mk(n, n) in higher dimensional spaces. Math. Compt. Simlation, 59: (2002), [3] A.M. Wazwaz. Exact special soltions with solitary patterns for the nonlinear dispersive K(m, n) eqations. Chaos, Solitons Fractals, 3: (2002), [4] A.M. Wazwaz. New solitary-wave special soltions with compact spport for the nonlinear dispersive K(m, n) eqations. Chaos, Solitons Fractals, 3 :(2002), [5] A. Biswas. -soliton soltion of the K(m, n) eqation with generalized evoltion. Phys. Lett. A., 372: (2008), [6] J. Yin, L. Tian. Peakon compacton soltions for K(p,q,) eqation. International Jornal of Nonlinear Science, 3 (2) :(2007), [7] Philip Rosena, Doron Levy. Compactons in a class of nonlinearly qintic eqations. Phys. Lett. A. 252: (999), [8] Philip Rosena, James M. Hyman, Martin Staley.Mltidimensional compactons. Phys. Rev. Lett., 98:(2007),-4. [9] T.R. Marchant. Asymptotic solitons of the extended Korteweg-de Vries eqation. Phys. Rev. E.,59: (999), IJNS for contribtion: editor@nonlinearscience.org.k

9 H. Triki, A. Wazwaz: Solitary Wave Soltions for a K(m,n,p,q+r) Eqation with Generalized Evoltion 395 [20] T.R. Marchant. Asymptotic solitons for a higher-order modified Korteweg-de Vries eqation. Phys. Rev. E. 66: (2002),-8. [2] A.M. Wazwaz. Exact soltions with compact noncompact strctres for the one-dimensional generalized Benjamin-Bona-Mahory eqation. Commn. Nonlinear Sci. Nmer. Siml., 0 :(2005), [22] F. Rs, F. R. Villatoro. Self-similar radiation from nmerical Rosenayman compactons. J. Compt. Phys., 227 :(2007), [23] T. Yoneyama. Interacting Korteweg-de Vries eqation attractive soliton interaction. Prog. Theoret. Phys., 72(6): (984), [24] R.Hirota, X. B. H, X. Y. Tang. A vector potential KdV eqation vector Ito eqation: soliton soltions, bilinear Backlnd transformation Lax Pairs. J.Math. Anal. Appl., 288: (2003), [25] R.Hirota. Discretization of copled modified KdV eqations. Chaos, Solitons Fractals, :(2000), [26] R. Hirota, M. Ito. Resonance of solitons in one dimension. J. Phys. Soc. Japan, 52(3): (983), [27] M. Ito. An extension of nonlinear evoltion eqations of the K-dV (mk-dv) type to higher order. J. Physical Society of Japan, 49(2): (980), [28] A.M.Wazwaz. Partial Differential Eqations Solitary Waves Theorem. Springer HEP, Berlin, (2009). [29] A.M.Wazwaz. Mltiple soliton soltions for copled KdV copled KP Systems.Canadian Jornal of Physics, 87( 2): (200), [30] A.M.Wazwaz. Mltiple front soltions for the Brgers eqation the copled Brgers eqations.appl. Math. Compt., 90: (2007), [3] A.M.Wazwaz. New solitons kink soltions for the Gardner eqation.commnications in Nonlinear Science Nmerical Simlation, 2(8): (2007), [32] M. Dehghan, A. Shokri. A nmerical method for soltion of the two-dimensional sine-gordon eqation sing the radial basis fnctions. Compt. Math. Simlation, 79 :(2008), [33] S.M.El-Sayed, D. Kaya. A nmerical soltion an exact explicit soltion of the NLS eqation. Appl. Math. Compt., 72 :(2006), IJNS homepage: