New Exact Solutions for MKdV-ZK Equation

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1 ISSN (print) (online) International Journal of Nonlinear Science Vol.8(2009) No.3pp New Exact Solutions for MKdV-ZK Equation Libo Yang 13 Dianchen Lu 1 Baojian Hong 2 Zengyong Huang 1 1 Nonlinear Scientific Research Center Jiangsu University Jiangsu P.R.China 2 Nanjing Institute of Technology Nanjing P.R.China 3 Huaiyin Institute of Technology Jiangsu P.R.China (Received 5 May 2009 accepted 21 September 2009) Abstract: In this paper we extended the Jacobian elliptic functions expansion method and gave the unified form expression of the Jacobian elliptic functions form solutions. With the aid of mathematica software we obtained abundant new exact solutions of mkdv-zk equation by using this method. These solutions are degenerated to the solitary wave solutions and the triangle function solutions in the cases when the modulus of the Jacobian elliptic functions degenerated to 1 and 0. Keywords: mkdv-zk equations; extended Jacobian elliptic functions expansion method; exact solutions; periodical solutions 1 Introduction In recent years the investigation of exact solutions to nonlinear evolution equations plays an important role in the nonlinear physical phenomena.several powerful methods have been proposed to construct exact solutions for nonlinear partial differential equations such as homogeneous balance method the hyperbolic function method the F-expansion method inverse scattering method projective Riccati equations method and so on [1-7]. By using these methods we obtained many valuable exact solutions for nonlinear evolution equations. In order to find more extended exact solutions for nonlinear evolution equations Liu presented the Jacobi elliptic functions expansion method [8-9]. Due to this method can be realized with the aid of computer algebra system it has an extensive promotion and application [10-11]. In this paper based on previous we extended the Jacobian elliptic functions expansion method and obtained a lot of new exact solutions of mkdv-zk equations by using this method. This paper is arranged as follows. In section 2 we briefly describe the new generalized Jacobian elliptic functions expansion method. In section 3 we obtain several families of solutions for mkdv-zk equation. In section 4 some conclusions are given. 2 Method For the given nonlinear evolution equations(nees) with two variables x and t where F is a polynomial with the variables u u t u x. We make the gauge transformation F (u u x u t u tt u xt u xx...) = 0 (1) u(x t) = u(ξ) ξ = k(x ct) (2) Supported by youth science Foundation of Huaiyin Institute of Technology(HGCO922) Corresponding author. address: liboyang80@126.com Copyright c World Academic Press World Academic Union IJNS /284

2 L. Yang D. Lu B. Hong Z. Huang: New Exact Solutions for MKdV-ZK Equation 319 where k and c are constants to be determined later. Substituting (2) into Eq.(1) yields a complex ordinary differential equation of u(ξ) namely We assume Eq. (3) has the following formal solutions : u(ξ) = a 0 + a i F i (ξ) + F ( u u u... ) = 0 (3) b i F i 1 (ξ)e(ξ) + c i F i 1 (ξ)g(ξ) + d i F i 1 (ξ)h(ξ) (4) where a 0 a i b i c i d i (i = 1 2 n) are constants to be determined later. ξ = ξ(x t) is arbitrary function with the variable x and t. The positive integer n can be determined by considering the homogeneous balance between governing nonlinear terms and the highest order derivatives of u in Eq. (3). And F (ξ) E(ξ) G(ξ) H(ξ) are an arbitrary array of the four functions e f g h. The selection obey the principle which make the calculation more simple. Here we ansatz e = e(ξ) = 1 f = f(ξ) = sn[ξ] (5) g = g(ξ) = cn[ξ] h = h(ξ) = dn[ξ] (6) where p j r s are arbitrary constants the four functions e f g h satisfy the following restricted relations e = jgh + rfh + sm 2 fg f = pgh + reh + seg g = pfh jeh + s(m 2 1)ef h = m 2 pfh r(m 2 1)ef jeg (7) g 2 = e 2 f 2 h 2 = e 2 m 2 f 2 where denotes d dξ m is the modulus of the Jacobian elliptic function (0 m 1). Substituting (4) and (7) into Eq.(3) yields a polynomial equation forf (ξ) E(ξ) G(ξ) H(ξ). Setting the coefficients of F i (ξ)e j (ξ)g k (ξ)h s (ξ)(i = )(j = 0 1)(k = 0 1)(s = 0 1) to zero yields a set of nonlinear algebraic equations(naes) ina 0 a i b i c i d i (i = n) and ξ solving the NAEs by Mathematica and Wu elimination we obtain a 0 a i b i c i d i (i = n). Substituting these results into(4)we can obtain the exact solutions of Eq.(1). Obviously the type (4) (5) and (6) are the general forms of the relevant expression in Ref. [9-12] if we choose special value of p j r and s we can get the results in Ref. [9-12]. 3 Exact solutions of the mkdv-zk equation Let us consider the mkdv-zk equations with following form: We introduce a gauge transformation Substituting (9) into (8) we have u t + qu 2 u x + u xxx + u xyy + u xzz = 0 (8) u = u(ξ) ξ = k(x ct + λy + ηz) (9) cu qu3 + +k 2 (1 + λ 2 + η 2 )u = 0 (10) By balancing the governing nonlinear terms and the highest order derivatives in Eq.(10) we obtain n = 1. Thus we assume Eq.(8) has the following solutions u(ξ) = a 0 + a 1 F (ξ) + b 1 E(ξ) + c 1 G(ξ) + d 1 H(ξ) (11) IJNS homepage:

3 320 International Journal of Nonlinear ScienceVol.8(2009)No.3pp Substituting (7) and (11) into Eq.(10) and setting the coefficients of F i (ξ)e j (ξ)g k (ξ)h s (ξ)(i = ) (j = 0 1)(k = 0 1)(s = 0 1) to zero yields nonlinear algebraic equations with respect to the unknowns k a 0 a 1 b 1 c 1 d 1. We could determine the following solutions: Family1: Whens = j = 0 we have e = 1 p rg then selectf (ξ) = g(ξ) E(ξ) = e(ξ) G(ξ) = f(ξ) H(ξ) = h(ξ). Case 1: a 1 = b 1 = c 1 = 0 d 1 = ±p q(1 + 2m 4 3m 2 ) r = εmp m 2 1 k 1 = ± 2c (1 2m 2 )(1 + η 2 + λ 2 ) Case 2: a 1 = b 1 = 0 c 1 = ±mp d 1 = εp r = 0 k 2 = ± 2c (1 + m 2 )(1 + η 2 + λ 2 ) Case 3: Case 4: c 1 = b 1 = 0 a 1 = ± (r 2 + m 2 (p 2 r 2 )) d 1 = ε (p 2 r 2 ) q(1 2m 2 ) q(1 2m 2 ) k 3 = k 1 p b 1 = c 1 = d 1 = 0 a 1 = ± q(2m 2 1) r = εp k 4 = k 1 Case 5: b 1 = d 1 = 0 a 1 = ±mp q(2 m 2 ) c 1 = mp q(m 2 2) r = 0 k 5 = ± 2c (m 2 2)(1 + η 2 + λ 2 ) where ε = ±1 i = 1 a 0 = 0 and p is a constant and not equal to zero. Therefore we obtain the following solutions to the Eq.(8). dnξ u 1 = ± 1 q(2m 2 1) m εmcnξ 1 u 3 = u 2 = q(m 2 + 1) ( mcnξ 2 + εdnξ 2 ) ± r 2 + m 2 (p 2 r 2 )snξ 3 + ε r 2 p 2 dnξ 3 q(2m 2 1) p + rcnξ 3 u 4 = snξ 4 u 5 = m (± cnξ 5 q(2m 2 1) 1 + εcnξ 4 q m εsnξ 5 ) 2 m 2 where ξ i = k i (x ct + λy + ηz) (i = ) c λ η are arbitrary constants. Family2: When r = j = 0 we have h = 1 s pe then selectf (ξ) = e(ξ) E(ξ) = f(ξ) G(ξ) = g(ξ) H(ξ) = h(ξ). Case 6: b 1 = c 1 = d 1 = 0 a 1 = ±m2 p q(2 m 2 ) s = εp k 6 = ± 2c (m 2 2)(1 + η 2 + λ 2 ) Case 7: a 1 = b 1 = d 1 = 0 c 1 = ±m 2 p q(2 3m 2 + m 4 ) s = εp 1 m 2 k 7 = k 6 Case 8: b 1 = d 1 = 0 a 1 = ±m (p 2 + (m 2 1)s 2 ) q(m 2 2) c 1 = εm (p 2 s 2 ) k 8 = k 6 q(m 2 2) IJNS for contribution: editor@nonlinearscience.org.uk

4 L. Yang D. Lu B. Hong Z. Huang: New Exact Solutions for MKdV-ZK Equation 321 whereε = ±1 i = 1 a 0 = 0and pis a constant and not equal to zero. Therefore we obtain the following solutions to the Eq.(8). u 6 = ±m2 snξ 6 u 7 = ±m2 cnξ 7 q(2 m 2 ) 1 + εdnξ 6 q(2 m 2 ) 1 m 2 + εdnξ 7 u 8 = m ± p 2 + (m 2 1)snξ 8 + ε s 2 p 2 cnξ 8 q(2 m 2 ) p + sdnξ 8 where ξ i = k i (x ct + λy + ηz) (i = 6 7 8) c λ η are arbitrary constants. Family3: When p = j = 0 we have h = 1 s rg then select F (ξ) = g(ξ) E(ξ) = e(ξ) G(ξ) = f(ξ) H(ξ) = h(ξ). Case 9: a 1 = c 1 = d 1 = 0 b 1 = ±(m2 1) r = εms k 9 = ± 2c (1 + m 2 )(1 + η 2 + λ 2 ) Case 10: c 1 = d 1 = 0 a 1 = ± (m 2 1)(r 2 m 2 s 2 ) b 1 = ε (m 2 1)(r 2 s 2 ) k 10 = k 9 Case 11: a 1 = d 1 = 0 b 1 = ±s (m 2 1) q(2m 2 1) c 1 = εms q(2m 2 1) r = 0 k 11 = ± 2c (2m 2 1)(1 + η 2 + λ 2 ) Case 12: a 1 = b 1 = 0 c 1 = ±s (1 + m 2 ) d 1 = εs r = εs 1 + m 2 k 12 = ik 9 2q q 2 where ε = ±1 i = 1 a 0 = 0 and s is a constant and not equal to zero. Therefore we obtain the following solutions to the Eq.(8). u 9 = ±(m2 1) 1 εmcnξ 9 + dnξ 9 u 10 = (m 2 1) ± r 2 m 2 s 2 snξ 10 + ε r 2 s 2 rcnξ 10 + sdnξ 10 u 11 = ± m εmcnξ 11 snξ 12 + εdnξ 12 u 12 = q(2m 2 1) dnξ 11 q ε 1 + m 2 + 2dnξ 12 whereξ i = k i (x ct + λy + ηz) (i = ) c λ η are arbitrary constants. Family4: When s = p = 0 we have g = 1 r jf then select F (ξ) = f(ξ) E(ξ) = e(ξ) G(ξ) = g(ξ) H(ξ) = h(ξ). Case 13: a 1 = b 1 = c 1 = 0 d 1 = ±rm q(2 m 2 ) j = εr m 2 1 k 13 = ± 2c (m 2 2)(1 + η 2 + λ 2 ) Case 14: b 1 = c 1 = 0 a 1 = ±r (1 m 2 ) d 1 = q(m 2 2) εr q(2m 2 1) j = 0 k 14 = ± 2c (1 2m 2 )(1 + η 2 + λ 2 ) IJNS homepage:

5 322 International Journal of Nonlinear ScienceVol.8(2009)No.3pp Case 15: a 1 = c 1 = 0 b 1 = ± (j 2 (m 2 1)r 2 ) d 1 = ε (j 2 + r 2 ) q(2 m 2 ) q(2 m 2 ) j = 0 k 15 = k 13 Case 16: a 1 = c 1 = d 1 = 0 b 1 = ±mr q(m 2 2) j = εir k 16 = k 13 where ε = ±1 i = 1 a 0 = 0 and r is a constant and not equal to zero. Therefore we obtain the following solutions to the Eq.(8). u 13 = ±m dnξ 13 q(2 m 2 ) ε m 2 1snξ 13 + cnξ 13 u 14 = ( ± 1 m 2 snξ 14 εdnξ + 14 ) q m 2 2cnξ 14 2m 2 1cnξ 14 u 15 = ± j 2 (m 2 1)r 2 + ε j 2 + r 2 dnξ 15 q(2 m 2 ) jsnξ 15 + rcnξ 15 u 16 = ±m 1 q(m 2 2) εisnξ 16 + cnξ 16 where ξ i = k i (x ct + λy + ηz) (i = ) c λ η are arbitrary constants. Family5: When s = r = 0we have e = 1 p jf then select F (ξ) = f(ξ) E(ξ) = e(ξ) G(ξ) = g(ξ) H(ξ) = h(ξ). Case 17: (m a 1 = b 1 = 0 c 1 = 2 p 2 j 2 ) d 1 = ε (p 2 j 2 ) k 17 = k 9 Case 18: Case 19: a 1 = b 1 = d 1 = 0 c 1 = ±p (m 2 1) j = εp k 18 = k 9 a 1 = b 1 = c 1 = 0 d 1 = ±p (1 m 2 ) q(2m 2 1) j = εmp k 19 = k 9 where ε = ±1 i = 1 a 0 = 0 and p is a constant and not equal to zero. Therefore we obtain the following solutions to the Eq.(8). u 17 = u 18 = ± (m 2 1) q(m 2 + 1) m ± m 2 p 2 j 2 cnξ 17 + ε p 2 j 2 dnξ 17 p + jsnξ 17 (1 m 2 ) cnξ 18 u 19 = ± 1 + εsnξ 18 q(2m 2 1) dnξ εmsnξ 19 where ξ i = k i (x ct + λy + ηz) (i = ) c λ η are arbitrary constants. Family6: When p = r = 0 we have h = 1 s jf then select F (ξ) = f(ξ) E(ξ) = e(ξ) G(ξ) = g(ξ) H(ξ) = h(ξ). Case 20: Case 21: b 1 = d 1 = c 1 = 0 a 1 = a 1 = d 1 = 0 b 1 = ±m2 s 6c q(2m 2 1) j = 0 k 20 = ±s 2q(1 2m 2 ) c 1 = c (2m 2 1)(1 + η 2 + λ 2 ) εs 2q(2m 2 1) j = εs 1 2m 2 2 k 21 = k 14 IJNS for contribution: editor@nonlinearscience.org.uk

6 L. Yang D. Lu B. Hong Z. Huang: New Exact Solutions for MKdV-ZK Equation 323 where ε = ±1 i = 1 a 0 = 0 and s is a constant and not equal to zero. Therefore we obtain the following solutions to the Eq.(8). u 20 = ±m2 6c snξ 20 u 21 = q(2m 2 1) dnξ 20 ±1 + εcnξ 21 q(1 2m 2 ) ε 1 2m 2 snξ dnξ 21 where ξ i = k i (x ct + λy + ηz) (i = 20 21) c λ η are arbitrary constants. When the module of the Jacobian elliptic function m 1 or m 0 these solutions degenerated to the solitary wave solutions and the triangle function solutions. 4 Conclusion By using this extended Jacabian elliptic functions method we obtained many new exact solutions of mkdv- ZK equation for example u 3 u 8 u 10 u 15 u 17. By using our method all results of Ref.[12] can be obtained. The method in this article contained traditional Jacobian elliptic functions method. This method applies to a wider range. By using this method we can get abundant solutions of nonlinear evolution equations. References [1] Lixia Wang Jiangbo Zhou Lihong Ren: The exact solitary wave solutions for family of BBM equation. International Journal of nonlinear science. 1(9):58-63 (2006) [2] Dianchen Lu Baojian Hong Lixin Tian: Explicit and exact solutions to the variable coefficient combined KdV equations with forced term. Acta Phys. Sin. 55(11): (2006) [3] Dianchen Lu Baojian Hong Lixin Tian: Backlund transformation and N-soliton-like solutionsto the combined kdv-burgers equation with variable coefficients. International Journal of nonlinear science. 2(1): 3-10 (2006) [4] Lixia Wang: New exact solutions for the MNV Equation. International Journal of nonlinear science. 5(2): (2008) [5] Engui Fan: Extended tanh-function method and its application to nonlinear equations. Phys.Lett. A. 277: (2000) [6] Dingjiang Huang Hongqing Zhang: Extended hyperbolic function method and new exact solitary wave solutions of Zakharov equations. Acta Phys.Sin. 53(8): (2004) (in Chinese) [7] Baojian Hong Dianchen Lu: New exact solutions for the generalized BBM and Burgers-BBM equations. World Journal of Modelling and Simulation. 4(4): (2008) [8] Liu Shishi Fu Zuntao Liu Shida et al: Expansion method about the Jacobi elliptic function andits applications to nonlinear wave equations. Acta Phys. Sin. 50(11): (2001) (in Chinese) [9] Fu Zuntao Liu Shishi Liu Shida et al: New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations Phys.Lett.A. 290(1-2):72-76 (2001) [10] Baojian Hong Dianchen: New Jacobi Elliptic function solutions for the Higher-order nonlinear Schrodinger Equation. International Journal of nonlinear science. 7(3): (2009) [11] Jiahua Han Junmao Wang Guojiang Wu et al: The extended expansion method for Jacobi elliptic function and the new exact periodic solutions for Klein-Gordon. Journal of Anhui University Natural Science Edition. 32(1):44-48 (2008) [12] Miao Zhang Wenliang Zhang Junmao Wang et al: The new exact periodic solutions of BBM and mkdv-zk equations. Journal of Anhui University Natural Science Edition. 32(3):52-55 (2008) IJNS homepage: