New Jacobi Elliptic Function Solutions for Coupled KdV-mKdV Equation Yunjie Yang Yan He Aifang Feng Abstract A generalized G /G-expansion method is used to search for the exact traveling wave solutions of the coupled KdV-mKdV equation As a result some new Jacobi elliptic function solutions are obtained It is shown that the method is straightforward concise effective and can be used for many other nonlinear evolution equations in mathematical physics Index Terms generalized G /G-expansion method; the coupled KdV-mKdV equation; Jacobi elliptic function solutions I INTRODUCTION Seeking the traveling wave solutions of nonlinear evolution equationsnlees has been an interesting and hot topic in mathematics physics for a long time Many effective methods to construct traveling wave solutions of NLEEs have been established [-] However no method can be used for finding all solutions for all types of NLEEs Recently the G /G-expansion method [] has become popular in the research community and the initial idea has been refined by many studies [3-7] It is shown that the G /G-expansion method is very effective and many nonlinear equations have been successfully solved In this paper some exact solutions of the coupled KdVmKdV equation which are expressed by the Jacobi elliptic function are obtained by using the G /G-expansion method A Description of the generalized G /G-expansion method Assume that the nonlinear partial differential equation F u u x u t u xx u xt u tt = 0 where F is a polynomial in its arguments The main steps of the generalized G /G-expansion method are descripted as follows Step Seeking traveling wave solutions of by taking ux t = uξ ξ = x ct and transforming to the ordinary differential equationode F u u cu c u cu u = 0 Step Looking for its solution uξ in the polynomial form m f i uξ = a 0 + a i 3 f i= This work was supported by the Natural Science Foundations of Education Committee of Yunnan Province 03C079 0Y45 Yunjie Yang is with the Department of Mathematics Kunming University Kunming Yunnan 6504 PR China e-mail: dfdyyj@63com Yan He is with the Department of Computer Science and Technology Chuxiong Teachers College Chuxiong Yunnan 675000 PR China Aifang Feng is with the Department of Mathematics Kunming University Kunming Yunnan 6504 PR China where a 0 a i i = m are constants which will be determined later f = fξ is the solution of the auxiliary LODE f = P f 4 + Qf + R 4 where P Q and R are constants Step 3 Determining the parameter m by balancing the highest order nonlinear term and the highest order partial derivative of u in Step 4 Substituting 3 and 4 into and setting all the coefficients of all terms with the same powers of f /f k k = to zero Then a system of nonlinear algebraic equations NAEs with respect to the parameters c a 0 a i i = m is obtained By solving the NAEs if available those parameters can be determined explicitly Step 5 Assuming that the constants c a 0 a i i = m can be obtained by solving the algebraic equations in Step 4 and substituting these constants and the known general solutions into 3 Then the explicit solutions of can be obtained immediately B Applications of method In this section we apply the G /G-expansion method to seek the exact solutions of the coupled KdV-mKdV equation [89] as follows: u t + uu x + βu u x + u xxx = 0 5 where and β are two constant parameters Let ux t = uξ ξ = x ct in 5 Then cu + uu + βu u + u = 0 6 where c is a constant which will be determined later By integrating both sides of 6 with respect to ξ cu + u + β 3 u3 + u = 0 7 Then m = by balancing u 3 and u in 7 According to 3 and 4 we have f uξ = a 0 + a 8 f u f ξ = a f Q 9 f f With the aid of Maple substituting 8 and 9 into 7 the left-hand side of 7 becomes a polynomial in f /f and ξ Setting their coefficients to zero yields a system of algebraic equations in a 0 a c β Solving these overdetermined algebraic equations we get the following result:
a 0 = Q a = Q c = 4Q β = 4Q 0 With the aid of the appendix [0] and from the formal solution 0 we get the following set of exact solutions of 5 Case Choosing P = m Q = + m R = and fξ = snξ we obtain the Jacobi elliptic function solution u = + m m csξdnξ ξ = x + 4 + m t Case Choosing P = m Q = + m R = and fξ = cdξ we obtain the Jacobi elliptic function solution u = m m sdξncξ + m ξ = x + 4 + m t Case 3 Choosing P = m Q = m R = m and fξ = cnξ we obtain the Jacobi elliptic function solution u 3 = m m dcξsnξ ξ = x 4m t 3 Case 4 Choosing P = Q = m R = m and fξ = dnξ we obtain the Jacobi elliptic function solution u 4 = m m m cdξsnξ ξ = x 4 m t 4 Case 5 Choosing P = Q = + m R = m and fξ = nsξ we obtain the Jacobi elliptic function solution u 5 = + m + m csξdnξ ξ = x + 4 + m t 5 Case 6 Choosing P = Q = + m R = m and fξ = dcξ we obtain the Jacobi elliptic function solution u 6 = m m scξndξ + m ξ = x + 4 + m t 6 Case 7 Choosing P = m Q = m R = m and fξ = ncξ we obtain the Jacobi elliptic function solution u 7 = m + m dcξsnξ ξ = x 4m t 7 Case 8 Choosing P = m Q = m R = and fξ = ndξ we obtain the Jacobi elliptic function solution u 8 = m + m m cdξsnξ ξ = x 4 m t 8 Case 9 Choosing P = m Q = m R = and fξ = scξ we obtain the Jacobi elliptic function solution u 9 = m + m dcξnsξ ξ = x 4 m t 9 Case 0 Choosing P = m m Q = m R = and fξ = sdξ we obtain the Jacobi elliptic function solution u 0 = m + m cdξnsξ ξ = x 4m t 0 Case Choosing P = Q = m R = m and fξ = csξ we obtain the Jacobi elliptic function solution u = m + m dsξncξ ξ = x 4 m t Case Choosing P = Q = m R = m m and fξ = dsξ we obtain the Jacobi elliptic function solution u = m m csξndξ ξ = x 4m t Case 3 Choosing P = /4 Q = m / R = /4 and fξ = nsξ ± csξ we obtain the Jacobi elliptic function solution u 3 = ξ = x m t m m dsξ 3 Case 4 Choosing P = m /4 Q = + m / R = m /4 and fξ = ncξ±scξ we obtain the Jacobi elliptic function solution u 4 = ξ = x + m t + m + + m dcξ 4 Case 5 Choosing P = /4 Q = m / R = m /4 and fξ = nsξ ± dsξ we obtain the Jacobi elliptic function solution u 5 = ξ = x m t m m csξ 5 Case 6 Choosing P = m /4 Q = m / R = m /4 and fξ = snξ ± icnξ we obtain the Jacobi elliptic function solution u 6 = + ξ = x m t m dnξcnξ isnξ snξ+icnξ m 6
Case 7 Choosing P = m /4 Q = m / R = m /4 and fξ = m sdξ ± cdξ we obtain the Jacobi elliptic function solution m cnξ+m snξ dnξ m snξ+cnξ u 7 = m + ξ = x m t m 7 Case 8 Choosing P = /4 Q = m / R = /4 and fξ = mcdξ ± i m ndξ we obtain the Jacobi elliptic function solution u 8 = m + m msnξ +m +i m mcnξ dnξmcnξ+i m ξ = x m t 8 Case 9 Choosing P = /4 Q = m / R = /4 and fξ = msnξ ± idnξndξ we obtain the Jacobi elliptic function solution m mcnξdnξ imsnξ msnξ+idnξ u 9 = m + ξ = x m t 9 Case 0 Choosing P = /4 Q = m / R = /4 and fξ = m scξ ± idcξ we obtain the Jacobi elliptic function solution m m dnξ isnξm cnξ m snξ+idnξ u 0 = m + ξ = x m t 30 Case Choosing P = m /4 Q = m + / R = m /4 and fξ = msdξ ± ndξ we obtain the Jacobi elliptic function solution u = ξ = x + m t + m + m + m cdξ 3 Case Choosing P = m /4 Q = m / R = /4 and fξ = snξ/ ± dnξ we obtain the Jacobi elliptic function solution u = ξ = x m t m + m csξ 3 Case 3 Choosing P = /4 Q = m + / R = m /4 and fξ = mcnξ ± dnξ we obtain the Jacobi elliptic function solution u 3 = m + m ξ = x m + t m + snξ 33 Case 4 Choosing P = m /4 Q = m + / R = /4 and fξ = dsξ ± csξ we obtain the Jacobi elliptic function solution u 4 = ξ = x m + t m + m + nsξ 34 Case 5 Choosing P = /4 Q = m / R = m /4 and fξ = dcξ ± m ncξ we obtain the Jacobi elliptic function solution u 5 = m + m snξ m + m dnξ cnξdnξ+ m ξ = x m t 35 Case 6 Choosing R = m Q / m + P Q < 0 P > 0 and fξ = m Q/m + P sn /m + ξ we obtain the Jacobi elliptic function solution u 6 = Q + Q +m cs +m ξ dn 36 ξ = x 4Qt +m ξ Case 7 Choosing R = m Q / m P Q > 0 P < 0 and fξ = / m P Q/ dn m ξ we obtain the Jacobi elliptic function solution u 7 = Q Q sn m ξ ξ = x 4Qt m m cd m ξ 37 Case 8 Choosing R = m m Q / m P Q > 0 P < 0 and fξ = Q/m m Q/m P cn ξ we obtain the Jacobi elliptic function solution u 8 = Q Q Q m dc Q m ξ Q sn m ξ ξ = x 4Qt 38 Case 9 Choosing P = Q = 4m R = and fξ = snξdnξ/cnξ we obtain the Jacobi elliptic function solution u 9 = 4m m sn ξ+m sn 4 ξ+ + 4m ξ = x 8 m t cnξsnξdnξ 39 Case 30 Choosing P = m 4 Q = m 4 R = and fξ = snξcnξ/dnξ we obtain the Jacobi elliptic function solution u 30 = m 4m sn 4 ξ sn ξ+ + m 4 ξ = x 8m t dnξsnξcnξ 40
Case 3 Choosing P = Q = m + R = m +m 4 and fξ = cnξdnξ/snξ we obtain the Jacobi elliptic function solution u 3 = m + + ξ = x 8m + t m +m sn 4 ξ dnξsnξcnξ 4 Case 3 Choosing P = A m /4 Q = m + / + 3m R = m /4A and fξ = cnξdnξ/a + snξ + msnξ we obtain the Jacobi elliptic function solution u 3 = m sn ξ+msn ξ m dnξcnξ m + + 3m + m + + 3m ξ = x m + + 6mt 4 Case 33 Choosing P = A m + /4 Q = m + / 3m R = m + /4A and fξ = cnξdnξ/a + snξ msnξ we obtain the Jacobi elliptic function solution u 33 = m sn ξ msn ξ+m dnξcnξ m + 3m + m + 3m ξ = x m + 6mt 43 Case 34 Choosing P = 4/m Q = 6m m R = m 3 + m 4 + m and fξ = mcnξdnξ/ msn ξ + we obtain the Jacobi elliptic function solution u 34 = snξ msn ξm+ mm+ dnξcnξmsn ξ+ 6m m + 6m m ξ = x 46m m t 44 Case 35 Choosing P = 4/m Q = 6m m R = m 3 + m 4 + m and fξ = mcnξdnξ/ msn ξ we obtain the Jacobi elliptic function solution u 35 = snξ msn ξm +mm + dnξcnξmsn ξ 6m m + 6m m ξ = x + 46m + m + t 45 Case 36 Choosing P = m + m + B Q = m + R = m m /B and fξ = msn ξ /B msn ξ + we obtain the Jacobi elliptic function solution u 36 = + m + ξ = x 8m + t 4m m +snξcnξdnξ m sn 4 ξ 46 Case 37 Choosing P = m m + B Q = m + R = m + m + /B and fξ = msn ξ + /B msn ξ we obtain the Jacobi elliptic function solution u 37 = + m + ξ = x 8m + t 4m m +snξcnξdnξ m sn 4 ξ 47 Remark The validity of all the solutions which are obtained are verified Remark In fact there are more than three solutions compared to the latest related works [] II CONCLUSION In this paper some exact solutions of Jacobi elliptic function form from the coupled KdV-mKdV equation are derived When the modulus of the Jacobi elliptic function m 0 or the corresponding solitary wave solutions and trigonometric function solutions are also obtained It is shown that the G /G-expansion method provides a very effective and powerful tool for solving nonlinear equations in mathematical physics REFERENCES [] CT Yan A simple transformation for nonlinear waves Phys Lett A 4 996 77-84 [] ML wang YB Zhou ZB Li Application of a homogeneous balance method to exact solutions of nonlinear evolution equations in mathematical physics Phys Lett A 6 996 67-75 [3] S Zhang A generalized auxiliary equation method and its application to +-dimensional Korteweg-de Vries equations Comput Math Appl 54 007 08-038 [4] EG Fan Extended tanh-function method and its applications to nonlinear equations Phys Lett A 77 000-8 [5] SK Liu ZT Fu SD Liu Q Zhao Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations Phys Lett A 89 00 69-74 [6] ZT Fu SK Liu SD Liu Q Zhao New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations Phys Lett A 90 00 7-76 [7] K Mohammed New Exact traveling wave solutions of the 3+- dimensional Kadomtsev-Petviaghvili KP equation IAENG International Journal of Applied Mathematics Vol 37 no pp 7-9 Aug 007 [8] T Hasuike Exact and Explicit Solution Algorithm for Linear Programming Problem with a Second-Order Cone IAENG International Journal of Applied Mathematics Vol 4 no 3 pp -5 Aug 0 [9] Y Warnapala R Siegel and J Pleskunas The Numerical Solution of the Exterior Boundary Value Problems for the Helmholtz s Equation for the Pseudosphere IAENG International Journal of Applied Mathematics Vol 4 no pp -6 May 0 [0] J V Lambers Solution of Time-Dependent PDF Through Componentwise Approximation of Matrix Functions IAENG International Journal of Applied Mathematics Vol 4 no pp -0 Feb 0 [] ML Wang XZ Li Application of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation Chaos Solitons and Fractals 4 005 57-68 [] ML Wang XZ Li and JL Zhang The G /G-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics Phys Lett A 37 008 47-43 [3] EME Zayed New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized G /G-expansion method Journal of Physics A Vol 4 no 9 Article ID 950 p 3 009 [4] SM Guo and YB Zhou The extended G /G-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations Applied Mathematics and Computation Vol 5 no 9 pp 34-3 00 [5] LX Li EQ Li and ML Wang The G /G /G-expansion method and its application to traveling wave solutions of the Zakharov equations Applied Mathematics Vol 5 no 4 pp 454-46 00
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