Similarity Solutions for Generalized Variable Coefficients Zakharov Kuznetsov Equation under Some Integrability Conditions

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1 Commun. Theor. Phys. Beijing, China) ) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 4, October 15, 010 Similarity Solutions for Generalized Variable Coefficients Zakharov Kuznetsov Equation under Some Integrability Conditions M.H.M. Moussa and Rehab M. El-Shiekh Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Hiliopolis, Cairo, Egypt Received January 14, 010) Abstract In this paper, the symmetry method has been carried over to the generalized variable coefficients Zakharov Kuznetsov equation. The infinitesimal symmetries and the optimal system are deduced and from this optimal system seven basic fields are determined, and for every vector field in the optimal system the admissible forms of the coefficients are found and this also leads us to transform the given equation into partial differential equations in two variables. After using some referenced transformations the mentioned partial differential equations eventually reduce to ordinary differential equations. The search for solutions to those equations has yielded many exact solutions in most cases. PACS numbers: 0.30.Jr Key words: symmetry method, the generalized variable coefficients Zakharov Kuznetsov equation, exact solutions 1 Introduction The generalized variable coefficients Zakharov Kuznetsov GVZK) equation is given by u t + αt)uu x + βt)u u x + ρt)u xx + λt)u xxx + γt)u xyy = 0, 1) where αt), βt), ρt), λt), and γt) are arbitrary functions of t. This equation contains many important equations for example when γt) = 0, this equation becomes the combined KdV-Burgers equation with variable coefficients [1] and when βt) = 0, and ρt) = 0 Eq. 1) turns to ZK equation with variable coefficients, which describe the nonlinear development of ion-acoustic waves in a magnetized plasma under the restrictions of small wave amplitude, weak dispersion, and strong magnetic fields [ 7] also when ρt) = 0, and λt) = 1 Eq. 1) turns to the generalized variable coefficients Zakharov Kuznetsov equation. [8 9] For the constant coefficients version of Eq. 1) and special cases see Refs. [10 16]. Symmetry Method We briefly outlined Steinberg s similarity method of finding explicit solutions of both linear and non-linear partial differential equations. [17] The method based on finding the symmetries of the differential equations is as follows: Suppose that the differential operator L can be written in the form Lu) = p u Hu), ) tp where u = ut, x) and H may depend on t, x, u, and any derivative of u as long the derivative of u does not contain more than p 1), t derivatives. Consider the symmetry operator called infintesimal symmetry, which being quasilinear partial differential operator of first-order, has the form Su) = At, x, u) u n t + B i t, x, u) u +Ct, x, u). 3) x i i=1 Define the Frèchet derivative of Lu) by FL, u, v) = d dε Lu + εv) ε=0. 4) With these definitions in the mind we need to follow the following steps: i) Compute FL, u, v); ii) Compute FL, u, Su)); iii) Substitute Hu) for p u/ t p ) in FL, u, Su)); iv) Set this expression to zero and perform a polynomial expansion; v) Solve the resulting partial differential equations. Once this system of partial differential equations is solved for the coefficients of Su), Eq. ) can be used to obtain the functional form of the solutions. 3 Fundamental Equation The generalized variable coefficients Zakharov Kuznetsov equation can be expressed in the form Lu) = u t + αt)uu x + βt)u u x + ρt)u xx + λt)u xxx + γt)u xyy = 0, 5) where αt), βt), ρt), λt), and γt) are arbitrary functions of t. 4 Determination of Symmetries In order to find the symmetries of Eq. 5), we set the following symmetry operator Su) = Ax, y, t, u)u t + Bx, y, t, u)u x + Cx, y, t, u)u y + Ex, y, t, u). 6) Calculating the Fréchet derivative FL, u, v) of Lu) in the direction of v, given by Eq. ), and replacing v by Su) in F, we get FL, u, Su)) = S t + αt)[us x + u x S] + βt)[u S x + uu x S] + ρt)s xx + λt)s xxx + γt)s xyy. 7) Substituting the values of different derivatives of Su) in F with the aid of Maple program, we get a polynomial expansion in u x, u t, u y, u x u t,..., etc. On making use of Eq. 5) in the polynomial expression for F, rearranging terms of various powers of derivatives of u and equating

2 604 M.H.M. Moussa and Rehab M. El-Shiekh Vol. 54 them to zero, we get A = At), B = Bt, x), C = Cy), E xu = E uu = 0, C yy + E yu = 0, ρb x Aρ t + 3λB xx A t ρ = 0, B t + αe + γe uyy + ρb xx + λb xxx + βue + βu B x +αub x Aβ) t u Aα) t u = 0, 3λB x Aλ) t = 0, γc y Aγ) t + γb x = 0, E t + αue x + βu E x + ρe xx + λe xxx + γe xyy = 0. 8) On solving system 8), we see that the infinitesimals A, B, C, and E satisfying the above equations are: A = 1 Γ t) [a 1 + a )Γt) + a 4 ], dγt) dt = αt), B = a 1 x + a 5, C = a 3 y + a 6, E = a u, 9) where a i, i = 1,,..., 6 are arbitrary constants. The functions αt), βt), ρt), λt), and γt) are governed by the relations: a 1+ a )β Aβ) t = 0, 3a 1 λ Aλ) t = 0, a 1 ρ Aρ) t = 0, a 1 + a 3 )γ Aγ) t = 0. 10) The symmetries under which Eq. 5) is invariant can be spanned by the following six infinitisimal generators V 1 = Γt) Γ t) t + x x, V = Γt) Γ t) t + u u, V 3 = y y, V 4 = 1 Γ t) t, V 5 = x, V 6 = y. 11) 5 Classification of Group-Invariant Solutions In general, to each s-parameter subgroup H of the full symmetry group G of a system of differential equations, there will correspond a family of group-invariant solutions. Since there are almost always an infinite number of such subgroups, it is not usually feasible to list all possible group-invariant solutions to the system. We need an effective, systematic means of classifying these solutions, leading to an optimal system of group-invariant solutions from which every other such solution can be derived. Since elements g G not in the subgroup H will transform an H-invariant solution to some other group-invariant solutions, only those solutions not so related need to be listed in our optimal system. Let G be a Lie group. An optimal system of s- parameter subgroups is a list of conjugacy inequivalent s-parameter subgroups with the proberty that any other subgroup is conjugate to precisely one subgroup in the list. The problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. For one-dimensional subalgebras, this classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation Olver 1986), [18] where the adjoint action is given by the Lie series Ad expεv i ))V j = V j ε[v i, V j ]+ ε [V i, [V i, V j ]]..., 1) where [V i, V j ] = V i V j V j V i is the commutator for the Lie algebra, and ε is a parameter. To obtain the optiaml system of the vector fields 11) we should first construct the commutator Table 1 as follows Table 1 The commutator table of the vector fields 11). V 1 V V 3 V 4 V 5 V 6 V V 4 V 5 0 V V V V 6 V 4 V 4 V V 5 V V V With the help of the Lie series 1) and the commutator table, the adjoint table for the Lie algebra 11) can be easily constructed as shown in Table. Table The adjoint table. V 1 V V 3 V 4 V 5 V 6 V 1 V 1 V V 3 expε)v 4 expε)v 5 V 6 V V 1 V V 3 expε)v 4 V 5 V 6 V 3 V 1 V V 3 V 4 V 5 expε)v 6 V 4 V 1 εv 4 V εv 4 V 3 V 4 V 5 V 6 V 5 V 1 εv 5 V V 3 V 4 V 5 V 6 V 6 V 1 V V 3 εv 6 V 4 V 5 V 6 To obtain the optiaml system, we now take a general element V = a 1 V 1 + a V + a 3 V 3 + a 4 V 4 + a 5 V 5 + a 6 V 6, 13) and subject it to various adjoint transformations to simplify it as much as possible; thus we have deduced the following basic fields which form an optimal system for the generalized variable coefficients Zakharov Kuznetsov equation i) V 1 + av + bv 3 ; ii) a) V + cv 3 + V 5, ii) b) V + cv 3 V 5 ; iii) V + wv 3 ; iv) a) V 3 + mv 4 + V 5, iv) b) V 3 + mv 4 V 5 ; v) V 3 + nv 4 ; vi) a) V 4 + kv 5 + V 6, vi) b) V 4 + kv 5 V 6 ; vii) V 4 + lv 5 ; viii) a) V 5 + V 6, viii) b) V 5 V 6 ; ix) V 5 ; x) V 6, where a, b, c, w, m, n, k, and l are arbitrary constants. Because the discrete symmetry x, y, t, u) x, y, t, u) will map ii) b), iv) b), vi) b), viii) b) to ii) a), iv) a), vi) a), viii) a) respectively, also the generators viii) a), ix), and x) give trivial cases since they do not depend on t, and therefore, in the optimal system, we confine ourselves to seven generators only. 6 Similarity Reductions and Reduced Ordinary Differential Equations In order to obtain the invariant transformation in each of the above cases we write the characteristic equation in the form dt Ax, y, t, u) = dx Bx, y, t, u) = dy Cx, y, t, u) du = Ex, y, t, u). 14) Once this equation is solved for the above seven cases the invariant variables and the corresponding reductions to partial differential equations are obtained and by using some reference transformations those partial differential

3 No. 4 Similarity Solutions for Generalized Variable Coefficients Zakharov Kuznetsov Equation under Some Integrability Conditions 605 equations will be reduced to ordinary differential equations under integrability conditions between the variable coefficients of the given problem. Our results are tabulated in the following Tables 3 5. Table 3 The invariant variables and the corresponding forms of the coefficient functions. Case The invariant variables ζ η u Forms of coefficient functions i) xγ 1/1+a) t) yγ b/1+a) t) fζ, η)γ a/1+a) t) βt) = k 1 Γ t)γ a/1+a) t), ρt) = k Γ t)γ 1 a)/1+a) t), λt) = k 3 Γ t)γ a)/1+a) t), γt) = k 4 Γ t)γ b a)/1+a) t). ii) ln Γt) x yγ c t) fζ, η)γ 1 t) βt) = k 5 Γt)Γ t), ρt) = k 6 Γ 1 t)γ t), λt) = k 7 Γ 1 t)γ t), γt) = k 8 Γ c 1 t)γ t). iii) x yγ w t) fζ, η)γ 1 t) βt) = k 9 Γt)Γ t), ρt) = k 10 Γ 1 t)γ t), λt) = k 11 Γ 1 t)γ t), γt) = k 1 Γ w 1 t)γ t). iv) x 1/m)Γt) y exp 1/m)Γt)) fζ, η) βt) = k 13 Γ t), ρt) = k 14 Γ t), λt) = k 15 Γ t), γt) = k 16 exp/m)γt))γ t). v) x y exp 1/n)Γt)) fζ, η) βt) = k 17 Γ t), ρt) = k 18 Γ t), λt) = k 19 Γ t), γt) = k 0 exp/n)γt))γ t). vi) x kγt) y Γt) fζ, η) βt) = k 1 Γ t), ρt) = k Γ t), λt) = k 3 Γ t), γt) = k 4 Γ t). vii) x lγt) y fζ, η) βt) = k 5 Γ t), ρt) = k 6 Γ t), λt) = k 7 Γ t), γt) = k 8 Γ t). Table 4 The corresponding reduced partial differential equation. k 4 f ζζζ + k f ζηη [1/1 + aζf ζ [b/1 + aηf η [a/1 + af + k 3 f ζζ + ff ζ + k 1 f f ζ = 0, k 7 f ζζζ + k 8 f ζηη + k 5 f f ζ + ff ζ f ζ + f k 6 f ζζ + cηf η = 0, k 11 f ζζζ + k 1 f ζηη + k 9 f f ζ + ff ζ f + k 10 f ζζ wηf η = 0, k 15 f ζζζ + k 16 f ζηη + k 13 f f ζ + ff ζ 1/m)f ζ + k 14 f ζζ 1/m)ηf η = 0, m 0 k 19 f ζζζ + k 0 f ζηη + k 17 f f ζ + ff ζ + k 18 f ζζ 1/n)ηf η = 0, n 0 k 3 f ζζζ + k 4 f ζηη + k 1 f f ζ + ff ζ + k f ζζ f η kf ζ = 0, k 7 f ζζζ + k 8 f ζηη + k 5 f f ζ + ff ζ + k 6 f ζζ lf ζ = 0, Table 5 Some referenced transformations to reduce the above partial differential equations to ordenary differential equations. Case The used transformation θ gθ) The ordinary differential equation i) c 1 ζ + c η f [1/1 + aθg [a/1 + ag + c 1 k g + c 1 gg + k 1 c 1 g g +k 4 c 1 c + k 3c 3 1 )g = 0, b = 1 ii) ζ + η f k 7 + k 8 )g + k 5 g g + gg g + g k 6 g = 0, c = 0 iii) c 1 ζ + c η f k 11 c k 1c 1 c )g + k 9 c 1 g g +c 1 gg g + c 1 k 10g = 0, w = 0 iv) c 1 ζ + c lnη f k 15 c 3 1 g + k 14 c 1 g + k 13 c 1 g g + c 1 gg 1/m)c 1 + c )g = 0, k 16 = 0 v) c 1 ζ + c lnη f k 19 c 3 1 g + k 18 c 1 g + k 17 c 1 g g +c 1 gg c /n)g = 0, k 0 = 0 vi) ζ + η f k 3 + k 4 )g + k 1 g g + gg 1 + k)g + k g = 0, vii) ζ + η f k 7 + k 8 )g + k 5 g g + gg l g + k 6 g = 0.

4 606 M.H.M. Moussa and Rehab M. El-Shiekh Vol Determination of Exact Solutions Now, we have going to our original task, finding exact solutions for the reduced ODEs, which by back substitution gives new exact solutions for the generalized variable coefficients Zakharov Kuznetsov equation as follows: Case i) To determine the solution for the ODE corresponding to this case, we assume that this solution takes the following form g = A 0 + A 1 θ + A θ + B 1 θ + B θ, 15) where A 0, A 1, A, B 1, and B are arbitrary constant to be determined. Substituting from Eq. 15) into the reduced ODE given by case i) and collecting the various powers of θ then equating them to zero, we get system of algebraic equations in the constants A 0, A 1, A, B 1, B, a, c 1, c, k 1, k, k 3, and k 4. Solving this system with the aid of Maple program, we get the following two sets of solutions The first set a = 1, A 1 = 1 c 1, B = 1k 3 c 1 + k 4c ), k 1 = k = A 0 = A = B 0 = B 1 = 0. The corresponding exact solution for Eq. 1) is given by ux, y, t) = x + c ) y Γ 1 t) c 1 1[k 3 + k 4 c /c 1 Γt) [x + c /c 1 )y]. 16) The second set a = 1, A 1 = 1 c 1, B 1 = k c 1, k 3 = k 4c c 1 k 1 = A 0 = A = B 0 = B = 0. Corresponding the second set we arrived at the following solution for the generalized variable coefficients Zakharov Kuznetsov equation ux, y, t) = x + c c 1 y ) Γ 1 t) + k x + c /c 1 )y. 17) Case ii) To obtain a solution for the ODE corresponding this case, we assume that k 7 = k 8 and k 5 = k 6 = 0, then we get the following form for g g = Lambert W C 0 expθ)), where C 0 is an integration constant. So that Eq. 1) has the following solution ux, y, t) = Γ 1 t) Lambert W C 0 expln Γt) x + y)). 18) Case iii) To solve the ODE corresponding to this case, we consider that k 10 = 0, and k 11 = c /c 1 )k 1, yields g = θ 1. c 1 k 9 k 9 Therefore the corresponding solution for the generalized variable coefficients Zakharov Kuznetsov equation [ ux, y, t) = x + c ) y 1 ] Γ 1 t). 19) k 9 c 1 k 9, Case iv) To find travelling wave solutions for the ODE corresponding to this case first of all integrate it with respect to θ, we got k 15 c 3 1g + k 14 c 1g + k 13c 1 3 g3 + c 1 g 1 m c 1 + c )g = 0, 0) where the integration constant equals zero. Now to solve Eq. 0) we have used the generalized tanh-function method given in Ref. [19] as follows. Let us assume that g takes the form s g = φ i ζ)a i, 1) i=0 where φζ) is a solution of the following Riccati equation φ ζ) = r + φ ζ). ) The previous equation has the following forms of solutions φζ) = r tanh rζ), r < 0, φζ) = r coth rζ), r < 0, φζ) = r tan rζ), r > 0, φζ) = r cot rζ), r > 0, φζ) = 1 ζ, r = 0. 3) Substituting from Eq. 1) into Eq. 0) and by making balance between the linear term g and the nonlinear term g 3 to determine the value of s, we have got that s = 1, so that g = A 0 + A 1 φ, 4) where A 0 and A 1 are arbitrary constants. Substituting Eqs. ) and 4) into Eq. 0) and setting to zero all coefficients of φ i i = 0, 1,, 3), we have obtained a system of algebraic equations, 1 3 k 13c 1 A k 14c 1 A 1r + 1 c 1A 0 c 1 + c )A 0 = 0, m k 13 c 1 A 1 A 0 + c 1A 0 A 1 + k 15 c 3 1 ra 1 c 1 + c )A 1 m k 15 c 3 1A k 13c 1 A 3 1 = 0, = 0, k 13 c 1 A 0 A 1 + k 14c 1 A c 1A 1 = 0. 5) Solving the above system with the aid of Maple program the constants A 0, A 1, k 13, k 14, k 15, c 1, c, r, and m are obtained as follows A 0 = 3, A 1 = 4c 1 k 14, k 15 = 8 3 k 13k 14, c = c 19m + 56k 13k 14c 1rm + 48k 13 ) 48k 13. 6) By substituting from Eq. 6) into Eq. 4) and by using the different forms of solutions of the Riccati equation ) which are given by Eq. 3), we arrive at the following exact solutions for Eq. 0)

5 No. 4 Similarity Solutions for Generalized Variable Coefficients Zakharov Kuznetsov Equation under Some Integrability Conditions 607 4c 1 k 14 r tanh[ rθ], r < 0, 4c 1 k 14 r coth[ rθ], r < 0, + 4c 1 k 14 r tan[ rθ], r > 0, 4c 1 k 14 r cot[ rθ], r > 0, 4c 1k 14, r = 0. θ Therefore by back substitution using the similarity variables corresponding to this case, we obtain the following exact solutions for Eq. 1) ux, y, t) = 4c 1 k 14 r tanh [ r c 1 x + c lny c 1 + c ) m Γt) 3, 7) ux, y, t) = 4c 1 k 14 r coth [ r c 1 x + c lny c 1 + c ) m Γt) 3, 8) + 4c 1 k 14 r tan [ r c 1 x + c lny c 1 + c ) m Γt), 9) [ r 4c 1 k 14 r cot c 1 x + c lny c 1 + c ) m Γt), 30) 4c 1 k 14 c 1 x + c ln y [c 1 + c )/m]γt). 31) Case v) To find solutions for the ODE corresponding to this case we used the generalized tanh-function method as we have already done in the previous case and we have got the following new exact solutions for the generalized variable coefficients Zakharov Kuznetsov equation [ r ux, y, t) = 4c 1 k 18 r tanh c 1 x + c lny c n Γt) 3, 3) where c, k 19 are given by ux, y, t) = 4c 1 k 18 r coth [ r c 1 x + c lny c n Γt) 3, 33) + 4c 1 k 18 r tan [ r c 1 x + c lny c n Γt), 34) 4c 1 k 18 r cot [ r c 1 x + c lny c n Γt), 35) 4c 1 k 18 c 1 x + c lny c /n)γt), 36) c = c 1n9 + 56k 17 k 18 c 1 r) 48k 17, k 19 = 8 3 k 17k 18, in the above solutions 3) 36). Case vi) Similarly exact solutions are found for the ODE corresponding this case via the generalized tanh-function method and therefore new classes of exact solutions to Eq. 1) are found to be [ r ux, y, t) = 4k r tanh x + y + 56k 1kr k 1 where Γt) 3, 4k 1 37) [ r ux, y, t) = 4k r coth x + y + 56k 1kr + 9 Γt) 3, 48k 1 4k 1 38) [ r + 4k r tan x + y + 56k 1kr + 9 Γt), 4k 1 48k 1 39) [ r 4k r cot x + y + 56k 1kr + 9 Γt), 4k 1 40) 48k 1 4k 4k 1 x + y + 3/16k 1 )Γt), 41) k = k 1 k r + 9 ), k 3 = k ) 48k 1 3 k 1k. Case vii) Corresponding the case under consideration, we repeat the above procedure by using the generalized

6 608 M.H.M. Moussa and Rehab M. El-Shiekh Vol. 54 tanh-function method, we arrive at the following sets of travelling wave solutions [ r ux, y, t) = 4k 6 r tanh x + y + 56k 5 k 6 r + 9 Γt) 3, 4) 48k 5 4k 5 [ r ux, y, t) = 4k 6 r coth x + y + 56k 5 k 6 r + 9 Γt) 3, 48k 5 4k 5 43) [ r + 4k 6 r tan x + y + 56k 5k6r + 9 Γt), 4k 5 48k 5 44) 4k 5 4k 6 r cot [ r x + y + 56k 5k6r k 5 Γt), 45) 4k 6 4k 1 x + y + 3/16k 5 )Γt). 46) References [1] D. Lu, B. Hong, and L. Tian, Int. J. Nonlinear Science 006) 3. [] E. Infeld, J. Plasma Phys ) 171. [3] E.W. Laedke and K.H. Spatschek, Phys. Rev. Lett ) 719. [4] E.W. Laedke and K.H. Spatschek, Phys. Fluids 5 198) 985. [5] E.W. Laedke and K.H. Spatschek, J. Plasma Phys ). [6] V.E. Zakharov and E.M. Kuznetsov, Sov. Phys. JTP ) 85. [7] Yan-Ze Peng, E.V. Krishnan, and Hui Feng, PRAMANA J. Phys ) 49. [8] S.A. El-Wakil, M.A. Madkour, and M.A. Abdou, Phys. Lett. A ) 6. [9] Z.L. Yan and X.Q. Liu, Appl. Math. and Comput ) 88. [10] Z.L. Yan and X.Q. Liu, Commun. Theor. Phys ) 9. [11] M.H.M. Moussa, Int. J. Eng. Sci ) [1] Kaya Dogan, Appl. Math. and Comput ) 79. [13] A.H. Khater, O.H. El. Kalaawy, and M.A. Helal, Choas, Solitons & Fractals ) [14] C.Z. Qu, Int. J. Theor. Phys ) 99. [15] B.K. Shivamoggi, J. Plasma Phys ) 83. [16] A.M. Wazwaz, Commun. Nonlinear Sci. Numer. Simul ) 597. [17] S. Steinberg, Symmetry Methods in Differential Equation, Technical Report, No. 367, University of New Mexico 1979). [18] P.J. Olver, Applications of Lie Group to Differential Equations, Springer Verlag, New York 1986). [19] E.G. Fan and Y.C. Hon, Z. Naturforsch 57a 00) 69.