JACOBI ELLIPTIC FUNCTION EXPANSION METHOD FOR THE MODIFIED KORTEWEG-DE VRIES-ZAKHAROV-KUZNETSOV AND THE HIROTA EQUATIONS

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1 JACOBI ELLIPTIC FUNCTION EXPANSION METHOD FOR THE MODIFIED KORTEWEG-DE VRIES-ZAKHAROV-KUZNETSOV AND THE HIROTA EQUATIONS ZAI-YUN ZHANG 1,2 1 School of Mathematics, Hunan Institute of Science Technology, College road, Yueyang, , Hunan Province, P. R. China 2 College of Science, National University of Defense Technology, Changsha, , Hunan, P. R. China zhangzaiyun1226@126.com Received November 2, 2014 In this paper, we investigate the modified Korteweg-de Vries-Zakharov-Kuznetsov the Hirota equations obtain exact travelling wave solutions by using Jacobi elliptic function expansion method together with the symbolic computation system Mathematica. For some limit cases, the obtained periodic solutions degenerate to the corresponding solitary wave solutions other types of solutions. Key words: modified Korteweg-de Vries-Zakharov-Kuznetsov equation, Hirota equation, Jacobi elliptic function expansion method, travelling wave solutions. PACS: Yv, Jr,42.81.Dp. 1. INTRODUCTION In recent years, nonlinear evolution equations (NEEs) in mathematical physics play a major role in various fields, such as fluid mechanics, plasma physics, optical fibers, so on. It is well known that the travelling wave solutions of NEEs play an important role in the study of nonlinear wave phenomena. Such wave phenomena are observed in fluid dynamics, plasma, elastic media, optical fibers, etc. (see for instance Ref. [1]). In the past decade, a lot of methods have been developed to construct travelling wave solutions to NEEs, such as the trigonometric function series method [1, 2], the modified mapping method the extended mapping method [3], the modified trigonometric function series method [4, 5], the dynamical system approach [6]-[8], the exp-function method [9], the multiple exp-function method [10], the transformed rational function method [11], the symmetry algebra method (consisting of Lie point symmetries) [12], the Wronskian technique [13], the modified ( G G G )-expansion method [14], the ( G 1 G )-expansion method [15, 16], the homogeneous balance method [17], the infinite series method the Jacobi elliptic function method [18, 19], the first integral method [20], the auxiliary ordinary differential equation method [21] so on. RJP Rom. 60(Nos. Journ. Phys., 9-10), Vol , Nos. 9-10, (2015) P , (c) 2015 Bucharest, - v.1.3a*

2 2 Jacobi elliptic function expansion method for mkdv-zk Hirota equations 1385 In this paper, we consider the (3+1)-dimensional modified Korteweg-de Vries- Zakharov-Kuznetsov (mkdvzk) equation [22] the Hirota equation [23] u t + βu 2 u x + u xxx + u xyy + u xzz = 0, (1) iu t + u xx + 2 u 2 u + i αu xxx + 6i α u 2 u x = 0, (2) we obtain exact travelling wave solutions, including the solutions of sn, cn, dn, cs type, by using the Jacobi elliptic function expansion method together with the symbolic computation system Mathematica. In some limit cases, these periodic solutions degenerate to the corresponding solitary wave solutions other types of solutions. Equation (1) governs the oblique propagation of electrostatic modes in magnetized plasmas has been (re)derived in a fully systematic way for general mixtures of hot isothermal, warm adiabatic fluid cold immobile background species [24]. Equation (2) describes the propagation of optical solitons in nonlinear optical fibers that exhibits a Kerr law optical nonlinearity [3, 4]. More recently, Baleanu et al. [23] established the different types of solutions of Eqs. (1) (2) by using the first integral method. Furthermore, they obtained the envelope, bell shaped, trigonometric, kink soliton solutions of these nonlinear evolution equations. It is well known that the generic nonlinear Schrödinger s equation (NLSE), was found to be completely integrable by the method of inverse scattering transformation (IST) [3, 4]. In the absence of the perturbation term, that is, when α = 0, Eq. (2) reduces to the NLSE with Kerr law nonlinearity iu t + u xx + α u 2 u = 0, (3) where α = 2. It is well known that the NLSE (3) admits the bright soliton solution, see Refs. [25] [4]: 2 u(x,t) = k α sech(k(x k 2 )t] 2µt))ei[µx (µ2, where µ k are arbitrary real constants, for the self-focusing case α > 0, the dark soliton solution, see Refs. [26] [4]: u(x,t) = k 2 α tanh(k(x 2µt))ei[µx (µ2 +2k 2 )t], where µ k are arbitrary real constants, for the self-defocusing case α < 0. Recently, Biswas Konar [27] investigated the optical solitons of Eq. (3). It is worth mentioning that Biswas et al. [28, 29] recently investigated the optical soliton perturbation with non-kerr law media iu t + u xx + F ( u 2 u) = iεr[u,u ].

3 1386 Zai-Yun Zhang 3 More details are presented in Refs. [28, 29]. It is worth mentioning that Zhang et al. [3], [4], [6], [7], [14] considered the NLSE with Kerr law nonlinearity iu t + u xx + α u 2 u + i[γ 1 u xxx + γ 2 u 2 u x + γ 3 ( u 2 ) x u] = 0 (4) obtained some new exact travelling wave solutions of Eq. (4). In Ref. [4], by using the modified trigonometric function series method, Zhang et al. studied some new exact travelling wave solutions. In Ref. [3], by using the modified mapping method the extended mapping method, it was derived some new exact solutions of Eq. (4), which are a linear combination of two different Jacobi elliptic functions it was investigated the solutions in some limit cases. In Ref. [6], by using the dynamical system approach, Zhang et al. obtained the travelling wave solutions in terms of bright dark optical solitons the cnoidal waves. It was found that the NLSE with Kerr-law nonlinearity has only three types of bounded travelling wave solutions, namely, bell-shaped solitary wave solutions, kink-shaped solitary wave solutions, Jacobi elliptic function periodic solutions. Moreover, it was pointed out the regions where these periodic wave solutions lie in. It was shown the relation between the bounded travelling wave solution the energy level h. It was observed that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it was shown that the exact periodic solutions evolve into solitary wave solutions. In Ref. [14], by using the modified ( G G )-expansion method, Miao Zhang obtained the travelling wave solutions of Eq. (4), which are expressed in terms of hyperbolic, trigonometric, rational functions. In Ref. [7], by using the theory of bifurcations, it was investigated the bifurcations dynamic behavior of travelling wave solutions to Eq. (4). Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions periodic wave solutions were obtained. 2. JACOBI ELLIPTIC FUNCTION EXPANSION METHOD AND EXACT TRAVELLING WAVE SOLUTIONS OF (1) AND (2) In this section, we investigate Eqs. (1) (2) given in Ref. [23] we obtain exact travelling wave solutions by using Jacobi elliptic function expansion method (JEFEM) [3], including the solutions of sn, cn, dn, cs type. In some limit cases, these periodic solutions degenerate to the corresponding solitary wave solutions other types of solutions. It follows from Ref. [23] that 3φ (ξ) βφ2 φ (ξ) λφ (ξ) = 0, ξ = x + y + z λt. (5)

4 4 Jacobi elliptic function expansion method for mkdv-zk Hirota equations 1387 Each side of Eq. (5) is integrated once 3φ βφ3 λφ = C 1, (6) where C 1 is a constant of integration. To facilitate further on our analysis, we assume that C 1 = 0, A = 3, B = λ, C = 1 3β. Then, Eq. (6) becomes the generic ordinary differential equation (ODE) Aφ + Bφ + Cφ 3 = 0. (7) On the other h, it is easy to see that Eqs. (2) also implies the ODE (7). In fact, benefiting from the ideas of Ref. [23], we make the transformation u = e iu φ(ξ), U = px + qt ξ = kx + ct. Substituting these transformations by separating real imaginary parts into Eq. (2) yields αk 2 φ + (c + 2pk 3αp 2 k)φ + 2αkφ 3 = C 1, (8) where C 1 is a constant of integration. We assume that C 1 = 0, A = αk 2, B = c + 2pk 3αp 2 k, C = 2αk. In what follows, we will discuss the travelling wave solutions to the ODE (7) JACOBI ELLIPTIC sn FUNCTION EXPANSION METHOD AND EXACT TRAVELLING WAVE SOLUTIONS OF (1) AND (2) To facilitate further on our analysis, we assume that Eq. (7) has the form n φ(ξ) = a j sn j (ξ), (9) j=0 where a j (j = 1,2,,n are constants the positive integer n can be determined by considering the balance of the highest order derivatives highest order nonlinear terms appearing in the ODE (7). So, we get n = 1. Eq. (9) reduces to We observe that [3] φ(ξ) = a 0 + a 1 snξ. (10) dφ dξ = a 1cnξdnξ, (11) d 2 φ dξ 2 = (1 + m2 )a 1 snξ + 2m 2 sn 3 ξ, (12) φ 3 (ξ) = a a 2 0a 1 snξ + 3a 0 a 2 1sn 2 ξ + a 3 1sn 3 ξ, (13) where m(0 < m < 1) is the modulus of the Jacobi elliptic function, see Ref. [3].

5 1388 Zai-Yun Zhang 5 It follows from Eqs. (7) (9)-(12) that That is, (B + Ca 2 0)a 0 + (B + 3Ca 2 0 (1 + m 2 )A)a 1 snξ + 3Ca 0 a 2 1sn 2 ξ +(Ca m 2 A)a 1 sn 3 ξ = 0, (B + Ca 2 0 )a 0 = 0, (B + 3Ca 2 0 (1 + m2 )A)a 1 = 0, 3Ca 0 a 2 1 = 0, (Ca m2 A)a 1 = 0. Solving the system (14) by Mathematica yields a 0 = 0, a 1 = ±m (14) C. (15) From (9) (14), we obtain φ(ξ) = ±m C sn B (x + y + z λt), (16) 1 + m2 where AC < 0 B < 0. For A = 3, B = λ, C = 1 3β we get φ(ξ) = ±m 18βsn λ (x + y + z λt), (17) 1 + m2 By using (9) (16), we get the travelling wave solutions of Eq. (1) as follows: u 11 (x,y,z,t) = m λ 18β sn (x + y + z λt). (18) 1 + m2 For A = αk 2, B = c + 2pk 3αp 2 k, C = 2αk we get φ(ξ) = ±m ksn c + 2pk 3αp2 k 1 + m 2 (kx + ct), (19) By using (9) (16), we get the travelling wave solutions of Eq. (2) as follows: u 21 (x,y,z,t) = m k sn c + 2pk 3αp2 k 1 + m 2 (kx + ct). (20) We notice that the Jacobi elliptic functions degenerate to the following functions (see, e.g. Ref. [3]) snξ tanhξ, as m 1. Taking m = 1, we have u 12 (x,y,z,t) = λ 18β tanh (x + y + z λt), (21) 2

6 6 Jacobi elliptic function expansion method for mkdv-zk Hirota equations 1389 u 22 (x,t) = k tanh c + 2pk 3αp2 k 1 + m 2 (kx + ct). (22) 2.2. JACOBI ELLIPTIC cn FUNCTION EXPANSION METHOD AND EXACT TRAVELLING WAVE SOLUTIONS OF (1) AND (2) We assume that Eq.(7) has the form n φ(ξ) = a j cn j (ξ), (23) j=0 where a j (j = 1,2,...,n are constants the positive integer n can be determined by considering the balance of the highest order derivatives the highest order nonlinear terms appearing in the ODE (7). So, we get n = 1. Then Eq. (22) reduces to φ(ξ) = a 0 + a 1 cnξ. (24) We observe that [3] dφ dξ = a 1snξdnξ, (25) d 2 φ dξ 2 = (2m2 1)a 1 cnξ 2m 2 a 1 cn 3 ξ, (26) φ 3 (ξ) = a a 2 0a 1 cnξ + 3a 0 a 2 1cn 2 ξ + a 3 1cn 3 ξ, (27) where m(0 < m < 1) is the modulus of the Jacobi elliptic function, see Ref. [3]. Substituting Eqs. (23)-(26) into Eq. (7) yields That is, (B + Ca 2 0)a 0 + (B + 3Ca (2m 2 1)A)a 1 cnξ + 3Ca 0 a 2 1cn 2 ξ +(Ca 2 1 2m 2 A)a 1 cn 3 ξ = 0, (B + Ca 2 0 )a 0 = 0, (B + 3Ca (2m2 1)A)a 1 = 0, 3Ca 0 a 2 1 = 0, (Ca 2 1 2m2 A)a 1 = 0. Solving the system (28) we get a 0 = 0, a 1 = ±m C. (29) From Eqs. (23) (28), we get φ(ξ) = ±m C cn B 2m 2 (x + y + z λt), (30) 1 (28)

7 1390 Zai-Yun Zhang 7 where AC > 0. By using (7) (29), we get the travelling wave solutions of Eqs. (1) (2) as follows: u 13 (x,y,z,t) = m λ 18β cn 2m 2 (x + y + z λt). (31) 1 u 23 (x,t) = m k cn c + 2pk 3αp2 k 2m 2 (kx + ct). (32) 1 We notice that the Jacobi elliptic functions degenerate to the following functions (see, e.g. Ref. [3]) cnξ sechξ, as m 1. Taking m = 1, we have u 14 (x,y,z,t) = 18β sech λ(x + y + z λt) (33) u 24 (x,t) = k sech (c + 2pk 3αp 2 k)(kx + ct). (34) 2.3. JACOBI ELLIPTIC dn FUNCTION EXPANSION METHOD AND EXACT TRAVELLING WAVE SOLUTIONS OF (1) AND (2) We assume that Eq. (7) has the form n φ(ξ) = a j dn j (ξ). (35) j=0 Similarly, we get n = 1. Then Eq. (35) reduces to We observe that φ(ξ) = a 0 + a 1 dnξ. (36) dφ dξ = m2 a 1 snξcnξ, (37) d 2 φ dξ 2 = (2 m2 )a 1 dnξ 2a 1 dn 3 ξ, (38) φ 3 (ξ) = a a 2 0a 1 dnξ + 3a 0 a 2 1dn 2 ξ + a 3 1dn 3 ξ, (39) where m(0 < m < 1) is the modulus of the Jacobi elliptic functions, see, e.g. Ref. [3]. Substituting Eqs. (35)-(39) into Eq. (7) yields (B + Ca 2 0)a 0 + (B + 3Ca (2 m 2 )A)a 1 dnξ + 3Ca 0 a 2 1dn 2 ξ +(Ca 2 1 )a 1 dn 3 ξ = 0,

8 8 Jacobi elliptic function expansion method for mkdv-zk Hirota equations 1391 That is, (B + Ca 2 0 )a 0 = 0, (B + 3Ca (2 m2 )A)a 1 = 0, 3Ca 0 a 2 1 = 0, (Ca 2 1 )a 1 = 0. Solving the system (40) by Mathematica gives (40) a 0 = 0, a 1 = ± C. (41) From (36) (40), we obtain φ(ξ) = ± C dn B (x ct), (42) 2 m2 where AC > 0 B < 0. By using (7) (42), we get the travelling wave solutions of Eq. (1) (2) as follows: u 15 (x,t) = 18β dn λ 2m 2 (x + y + z λt). (43) 1 u 25 (x,t) = kdn c + 2pk 3αp2 k 2 m 2 (kx + ct), (44) We notice that the Jacobi elliptic functions degenerate to the following functions (see Ref. [3]) Taking m = 1, we have u 16 (x,y,z,t) = dnξ sechξ, as m 1 C sech B(x + y + z λt) (45) u 26 (x,t) = k sech (c + 2pk 3αp 2 k)(kx + ct), (46) In fact, u 16 (x,y,z,t) is similar to u 14 (x,y,z,t) u 26 (x,t) is similar to u 24 (x,t) JACOBI ELLIPTIC cs FUNCTION EXPANSION METHOD AND EXACT TRAVELLING WAVE SOLUTIONS OF (1) AND (2) We assume that Eq. (7) has the form n φ(ξ) = a j dn j (ξ). (47) j=0

9 1392 Zai-Yun Zhang 9 Similarly, we get n = 1. Then Eq. (47) reduces to We observe that [3] φ(ξ) = a 0 + a 1 csξ, csξ = cnξ snξ. (48) dφ dξ = a 1(1 + cs 2 ξ)dnξ, (49) d 2 φ dξ 2 = (2 m2 )a 1 csξ + 2a 1 cs 3 ξ, (50) φ 3 (ξ) = a a 2 0a 1 csξ + 3a 0 a 2 1cs 2 ξ + a 3 1cs 3 ξ, (51) where m(0 < m < 1) is the modulus of the Jacobi elliptic functions, see e.g. Ref. [3]. Substituting Eqs. (47)-(49) into Eq. (7) yields That is, (B + Ca 2 0)a 0 + (B + 3Ca (2 m 2 )A)a 1 csξ + 3Ca 0 a 2 1cs 2 ξ +(Ca )a 1 cs 3 ξ = 0, (B + Ca 2 0 )a 0 = 0, (B + 3Ca (2 m2 )A)a 1 = 0, 3Ca 0 a 2 1 = 0, (Ca )a 1 = 0. Solving the system (52) by Mathematica gives a 0 = 0, a 1 = ± C. (53) From (47) (52), we obtain φ(ξ) = ± C cs B (x ct), (54) 2 m2 where AC < 0 B < 0. By (7) (54), we get the travelling wave solutions of Eqs. (1) (2) as follows: (52) u 17 (x,y,z,t) = 18β cs λ(x + y + z λt) (55) u 27 (x,y,z,t) = k cs (c + 2pk 3αp 2 k)(kx + ct). (56) We notice that the Jacobi elliptic functions degenerate to the following functions (see e.g. Ref. [3]) csξ cschξ, as m 1

10 10 Jacobi elliptic function expansion method for mkdv-zk Hirota equations 1393 Taking m = 1, we have u 18 (x,y,z,t) = 18β csch λ(x + y + z λt) (57) u 28 (x,y,z,t) = k csch (c + 2pk 3αp 2 k)(kx + ct). (58) 3. CONCLUSION AND DISCUSSION In this paper, we have investigated the travelling wave solutions of two nonlinear evolution equations that arise in mathematical physics problems we have reduced the problem to solving the following generic ordinary differential equation: Aφ (ξ) + Bφ(ξ) + Cφ 3 (ξ) = 0. We have then derived the exact travelling wave solutions of the corresponding nonlinear evolution equations by using Jacobi elliptic function expansion method. We have obtained travelling wave solutions of the sn, cn, dn, cs type. In particular cases, these periodic solutions degenerate to the corresponding solitary wave solutions other type of solutions. Finally, it is worthwhile to mention that this quite simple method can also be applied to solve other nonlinear evolution equations arising in mathematical physics problems. Further studies will be reported elsewhere. Acknowledgements. This work was supported by China Postdoctoral Science Foundation No. 2013M No. 2014T70991, the construct program of the key discipline in Hunan province No Science Technology Innovation Team in Colleges Universities in Hunan Province. REFERENCES 1. Z. Y. Zhang, Turk. J. Phys. 32, (2008). 2. W. X. Ma, B. Fuchssteiner, Int. J. Nonlinear Mechanics 31, (1996). 3. Z. Y. Zhang, Z. H. Liu, X. J. Miao, Y. Z. Chen, Appl. Math. Comput. 216, (2010). 4. Z. Y. Zhang, Y. X. Li, Z. H. Liu, X. J. Miao, Commun. Nonlinear Sci. Numer. Simulat. 16, (2011). 5. Z.Y. Zhang, Y.H. Zhang, X.Y. Gan, D.M. Yu, Zeitschrift für Naturforschung 67a, (2012). 6. Z. Y. Zhang, Z. H. Liu, X. J. Miao, Y. Z. Chen, Phys. Lett. A 375, (2011). 7. Z. Y. Zhang, X. Y. Gan, D. M. Yu, Zeitschrift für Naturforschung 66a, (2011). 8. Z. Y. Zhang, F. L. Xia, X. P. Li, Pramana 80, (2013). 9. A. Yıldırım, Z.Pınar, Comput. Math. Appl. 60, (2010). 10. W. X. Ma, T. W. Huang, Y. Zhang, Phys. Scr. 82, (2010). 11. W. X. Ma, J.-H. Lee, Choas, Solitons Fractals 42, (2009). 12. W. X. Ma, M. Chen, Appl. Math. Comput. 215, (2009). 13. W. X. Ma, Y. You, Trans. Amer. Math. Soc. 357, (2005). 14. X. J. Miao, Z. Y. Zhang, Commun. Nonlinear Sci. Numer. Simulat. 16, (2011).

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