New Exact Solutions of Two Nonlinear Physical Models

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1 Commun. Theor. Phys. Beijing China pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53 No. April 5 00 New Exact Solutions of Two Nonlinear Physical Models M.M. Hassan Mathematics Department Faculty of Science Minia University El-Minia Egypt Received May 3 009; revised manuscript received October Abstract Abundant new exact solutions of the Schamel Korteweg-de Vries S-KdV equation and modified Zakharov Kuznetsov equation arising in plasma and dust plasma are presented by using the extended mapping method and the availability of symbolic computation. These solutions include the Jacobi elliptic function solutions hyperbolic function solutions rational solutions and periodic wave solutions. In the limiting cases the solitary wave solutions are obtained and some known solutions are also recovered. PACS numbers: 0.30.Jr Fd 5.35.Sb 5.35.Fp Key words: modified Zakharov Kuznetsov equation Schamel Korteweg-de Vries equation traveling and periodic wave solutions extended mapping method Introduction Nonlinear partial differential equations NLPDEs are widely used as models to describe many important dynamical systems in various fields of sciences particularly in fluid mechanics solid state physics plasma physics and nonlinear optics. Exact solutions of NLPDEs of mathematical physics have attracted significant interest in the literature. The knowledge of these solutions of NLPDEs facilitates the verification of numerical solvers and aids in the stability analysis of the solutions. Traveling waves whether their solution expressions are in explicit or implicit forms are very interesting from the point of view of applications. Over the last years much work has been done on the construction of exact solitary wave solutions and periodic wave solutions of nonlinear physical equations. The investigation of new exact solutions of NLPDEs may help one to find new phenomena. Many methods have been developed by mathematicians and physicists to find special solutions of NLPDEs such as the inverse scattering method ] the tanh-function method ] the extended tanh-function method 3] Exp-function method ] sine-cosine method 5] and the homogeneous balance method. 6] Recently some methods were presented to construct exact solutions expressed in terms of Jacobi elliptic functions JEFs for nonlinear evolution equations NLEEs. Among them the Jacobi elliptic function expansion method 7 8] the F-expansion method 9 0] the generalized Jacobi elliptic function method ] mapping method ] extended mapping method 3 ] and other methods. 5 8] By employing a computer package such as Maple or Mathematica the large amount of tedious working required to verify candidate traveling wave solutions can be avoided. Hence these methods are now much more attractive and efficient for obtaining the exact solutions of some NLEEs. Some of the nonlinear models in fluids plasma and dust plasma are described by canonical models and include the Korteweg-de Vries KdV the modified KdV Zakharov Kuznetsov ZK and modified ZK equations. 9 5] Employing the reductive perturbation technique on the system of equations for hydrodynamics and the dynamics of plasma waves to derive the KdV equation the modified KdV the ZK equation and modified ZK equation. Schamel equation 3] containing a square root nonlinearity is a very attractive model for the study of ion-acoustic waves in plasmas and dusty plasmas and a simple solitary wave solution having a sech profile was obtained. Hence seeking new exact solutions of the Schamel Korteweg-de Vries S-KdV equation and the modified ZK equation is important. For a given NLPDE say in two independent variables Gu u t u x u tt u xx... = 0. In general the left hand side of Eq. is a polynomial in u and its various derivatives. We seek the traveling wave solution of in the form ux t = uξ; kx t where k and are constants to be determined later. Then Eq. is transformed to the ordinary differential equation ODE Hu u u... = 0 where u = du/dξ and H is a polynomial of u and its various derivatives. If H is not a polynomial of u and its various derivatives then we may use new variables v = vξ which makes H become a polynomial of v and its various derivatives. Let us give a brief description of the extended mapping method. 3 ] We assume that the solutions of Eq. can hassmamd@yahoo.com

2 No. New Exact Solutions of Two Nonlinear Physical Models 597 be expressed in the form ux t = u a N ai φ i ξ + b i φ i ξ i= N B i φ i ξφ ξ i= N i= c i φ i ξφ ξ 3 where N in Eq. 3 is a positive integer that can be determined by balancing the nonlinear term s with the highest derivative term in and a 0 a i b i B i and c i are constants to be determined. The function φξ satisfies the nonlinear ODE φ ξ = q 0 + q φ ξ + q φ ξ where q 0 q and q are constants. Substituting 3 with into the ODE and setting the coefficients of the different powers of φ i to zero yields a set of algebraic equations for a 0 a i b i B i c i k and. Solving the algebraic equations by use of Maple or Mathematica we have a 0 a i b i B i c i k and expressed by q 0 q q. Substituting the obtained coefficients into 3 then concentration formulas of traveling wave solutions of the NLPDE can be obtained. Selecting the values of q 0 q q and the corresponding JEFs φξ from Appendix and substituting them into the concentration formulas of solutions to obtain the explicit and exact JEF solutions of Eq.. Various solutions of Eq. were constructed using JEFs see Appendix and these results were exploited in the design of a procedure for generating solutions of NLPDEs. The JEFs = m cn cnξ m and = m where m 0 < m < is the modulus of the elliptic function are double periodic and posses the following properties: sn ξ + cn dn ξ + m sn d d = cnξ cn dξ dξ d dξ = m cnξ. In addition when m the functions cnξ and degenerate as tanhξ sechξ and sechξ respectively while when m 0 cnξ and degenerate as sinξ cosξ respectively. So we can obtain hyperbolic function solutions and trigonometric function solutions in the limit cases when m and m 0. Some more properties of JEFs can be found in 6]. The rest of this paper is organized as follows: In Sec. we apply the extended mapping method to S-KdV equation and drive the concentration formulas of solutions. Therefore we construct abundant new traveling wave solutions of S-KdV equation expressed by various JEFs. In Sec. 3 we find abundant exact solutions of modified ZK equation via this method. Solutions in the limiting cases are also obtained. A conclusion is given in Sec.. JEF Solutions of S-KdV Equation In this section we consider the S-KdV equation 9 0] u t + αu / + βuu x + δu xxx = 0 δβ 0 5 where α β and δ are constants. We refer to this equation as the Schamel Korteweg-de Vries S-KdV equation. This equation incorporates the KdV equation α = 0 and the Schamel equation β = 0. Due to the wide range of applications of equation 5 and the KdV equation many solutions to them have been obtained in terms of tanhξ sechξ and so on by using different methods. 7937] Beside these solutions there are several interesting types of solutions expressed by various JEFs and hyperbolic functions. Khater et al. 7] have used the F-expansion method and obtained the periodic wave solutions of Eq. 5. More recently Khater et al. 8] have obtained elliptic functions solutions of Schamel equation. Here we obtain several classes of exact solutions of S- KdV equation expressed by various JEFs and hyperbolic functions by using the extended mapping method and the availability of symbolic computation. In order to obtain the exact solutions of Eq. 5 we use the transformations ux t = v x t vx t = V ξ; kx t to reduce Eq. 5 to the ODE V V +αv +βv 3 V +δk V V +3V V ] = 0. 6 The balancing procedure implies that N =. Therefore we assume Eq. 5 has a solution in the form V x t = V a 0 + a φξ + b φξ + c φ ξ φξ 7 where a 0 a b and c are constants to be determined and φξ is a solution of Eq.. Substituting 7 into 6 and setting each coefficient to zero we get a system of nonlinear equations for a 0 a b c k and. Solving this system by use of Maple we get Case a 0 = α 5β a = ± α q b = 0 5β q c = 0 k = ± α 5 = 6α 6δβq 75β. 8 Case a 0 = α 5β a = 0 b = ± α q0 5β q c = 0 k = ± α 5 = 6α 6δβq 75β. 9

3 598 M.M. Hassan Vol. 53 Case 3 Case Case 5 a 0 = α 5β a = 0 b = 0 c = ± α 5β α k = ± q 5 = 6α 3δ βq 75β. 0 a 0 = α 5β a = ± α q 5β q 6 q 0 q b = ± α q 0 5β q 6 q 0 q α c = 0 k = ± 5 6 δβq 6 6 α = q 0 q 75 β a 0 = α 5β a = ± α q 5β q + 6 b = α q 0 q 0 q 5β q + 6 q 0 q α c = 0 k = ± 5 6δβq + 6 α = 6 q 0 q 75 β. a 0 = α 5β a = ± α q 5β q + 6 q 0 q b = ± α q 0 5β q + 6 q 0 q α c = 5 β q + 6 α k = ± q 0 q 5 3δβq + 6 q 0 q a 0 = α 5β a = ± α q 5β q 6 b = α q 0 q 5β = 6 α 75 β q 0 q 6 q 0 q 3 α c = 5 β q 6 α k = ± q 0 q 5 3δβq 6 6 α = q 0 q 75 β. Substituting 8 0 into 7 we obtain the following concentration formulas of traveling wave solutions of Eq. 5: u = α q ] α q0 ] 5β ± φξ u = q 5β ± 5 q φξ with α 5 x + 6α t and 6δβq 75β u = α 5β ± φ ξ ] α q φξ 5 x + 6α 3δ βq 75β t. 6 Moreover substituting into 7 we obtain the concentration formulas of solutions to Eq. 5 as follows: u = α 5β ± q 6 q φξ + ] α q 0 q 0 q φξ 5 6δβq 6 6α x + t 7 q 0 q 75 β u = α 5β ± q + 6 q φξ ] α q 0 q 0 q φξ 5 6δβq α x + t 8 q 0 q 75 β u = α q q0 5β ± q + 6 φξ + q 0 q φξ φ ξ ] α φξ 5 3δβq α x + q 0 q 75 β t 9 u = α q q0 5β ± q 6 φξ q 0 q φξ φ ξ ] α φξ 5 3δβq 6 6 α x +. 0 q 0 q 75 β With the aid of Appendix and formulas 5 one can get the periodic wave solutions of equation 5 u = α ] 5β ± m + m u = α ] 5β ± m + nsξ u 3 = α ] 5β ± m + m cdξ u = α ] 5β ± m + dcξ α with 5 6δβm + x + 6α 75β t βδ < 0 and u 5 = α 5β ± ] m u6 = α m ] 5β ± m ndξ 3

4 No. New Exact Solutions of Two Nonlinear Physical Models 599 α with x 5 + 6α 6δβ m 75β t βδ > 0. We also find some exact solutions of Eq. 5 expressed by rational expressions of JEFs u 7 = α 5β ± α 5 3δβm u 9 = α 5β ± α 5 3δβ + m u = α 5β ± u = α 5β ± and the combined JEF solutions m m ] u8 = α ± ] ± 5β ± m x + 6α 75β t βδ < 0 m + m + m m m m ± m ] + m ] u0 = α ± m 5β ± x + 6α 75β t βδ > 0 5 m ] α x m cnξ ± 5 + 6α 3δβ + m 75β t βδ > 0 ± i cnξ ] α 5 3δβm u 3 = α ] α 5β ± m cnξ ± + m u = α 5β ± m ± i cnξ m x + 6α 75β t βδ < 0 6 x 5 + 6α 3δβ + m 75β t βδ > 0 ] α 5 3δβm x + 6α 75β t βδ < 0. 7 From Appendix and the formulas 7 and 8 we obtain the following combined JEF solutions of Eq. 5: u 5 = α ] 5β ± + m + 6m m + nsξ u6 = α 5β ± α 5 6δβ + m + 6m u 7 = α 5β ± + m + 6m m cdξ + dcξ ] x + 6α 75β t βδ < 0 8 m + 6 m + m ndξ α 5 6δβ m + 6 m ] x + 6α 75β t δβ > 0. 9 With the aid of Appendix and the formulas 7 and 8 we find the following exact solutions of Eq. 5 expressed by rational expressions of JEFs: u 8 = α 5β ± u 9 = α 5β ± u 0 = α 5β ± u = α 5β ± m + m m + m + m m sn ξ + ± ] α x ± 5 + 6α 6δβm + 75β t δβ < 0 30 ± i cnξ + ] α x ± i cnξ 5 + 6α 6δβm + 75β t δβ < 0 3 sn ξ + ± cnξ ] α x + 6α 75β t βδ < 0 3 ± cnξ 5 6δβ + m mcnξ ± + m ] α m cnξ ± 5 6δβ m x + 6α 75β t βδ > By using the formulas 6 9 and 0 we can obtain new and more general types of exact solutions of 5. With the aid of Appendix and choosing φ in formulas 6 9 and 0 we can obtain the following exact

5 600 M.M. Hassan Vol. 53 solutions of Eq. 5 expressed by JEFs: u = α i 5β ± + m cnξ ] α 5 3δ β + m u 3 = α 5β ± m + nsξ cnξ ] 6m m α x 5 + 6α 3δβ6m m 75β t βδ < 0 u = α i 5β ± m nsξ cnξ ] + 6m + m α 5 3δβ + 6m + m x + 6α 75β t 3 x + 6α 75β t βδ > With the aid of Appendix and choosing φ / + in formulas 9 and 0 we can obtain u 5 = α m 5β ± m ] α csξ 5 3δβm u 6 = α 5β ± i m + m + + csξ ] α 5 3δβ + m x + 6α 75β t βδ < 0 x + 6α 75β t βδ > Other JEFs are omitted here for simplicity. In the limiting case when m the solitary wave solutions of Eq. 5 are obtained as follows: u 7 = α α ] 5β ± tanh x 5 + 6α 3δβ 75β t u8 = α α ] 5β ± coth x 5 + 6α 3δβ 75β t βδ < 0 37 u 9 = α 5β ± sech α ] x 5 + 6α 6δβ 75β t βδ > 0 38 u 30 = α 5β ± tanhξ ] u6 = α ± sechξ 5β ± ± sechξ ] α x tanhξ 5 + 6α 3δβ 75β t βδ < 0 39 ] u 3 = α 5β ± tanhξ + i sechξ u3 = α ] α 5β ± x tanhξ + i sechξ 5 + 6α 3δβ 75β t βδ < 0 0 u 33 = α 5β ± ] tanhξ + cothξ α x 5 + 6α δβ 75β t βδ < 0 u 3 = α 5β ± tanhξ + sechξ + + sechξ ] tanhξ cschξ α 5 3δβ x + 6α 75β t βδ < 0. The periodic wave solutions u u 6 were given in 7] and the solitary wave solutions u 7 and u 8 were found by many authors. 99] Compared with the results in 7] we further find some new solutions. As m 0 these solutions will degenerate into the corresponding trigonometric functions solutions. We do not list these solutions here for simplicity. 3 Solutions of Modified ZK Equation The equation is expressed as u t + αu u x + u xxx + u xyy = 0 where α is a constant. The ZK equation ] and modified ZK equation ] are very attractive models for the study in plasma. Exact traveling wave solutions for the ZK equation and modified ZK equation have been studied in ]. In order to obtain the exact solutions of modified ZK equation using the extended mapping method and the availability of symbolic computation we use the transformation ux y t = uξ kx + l y t with constants k l and to be determined. Then Eq. is transformed to the following ODE: u + α u u + k + l u = 0. 3 In Eq. 3 we assume uξ to be in the form u a 0 + a φξ + b φξ + c φ ξ φξ. Substituting this form into 3 and setting each coefficient to zero we obtain a system of nonlinear equations for a 0 a b c k and. Solving this system with the aid of Maple we get the following results: Case 6 q a 0 = b = c = 0 a = ± α q k = ± + l q.

6 No. New Exact Solutions of Two Nonlinear Physical Models 60 Case Case 3 a 0 = a = c = 0 b = ± a 0 = a = b = 0 c = ± Case 6 q a 0 = c = 0 a = ± α q 6 q 0 q b 6 q 0 = ± α q 6 q 0 q 6 q a 0 = c = 0 a = ± α q + 6 q 0 q b 6 q 0 = α q + 6 q 0 q 6 q0 k = ± α q + l. 5 q q k = ± α q + l. 6 q k = ± + l q 6 q 0 q 7 k = ± + l q + 6 q 0 q. 8 Case 5 q a 0 = 0 a = ± α q 6 q 0 q b q 0 = α q 6 q 0 q c = ± α q 6 q 0 q k = ± + l q 6 q 0 q 9 q a 0 = 0 a = ± α q + 6 q 0 q b q 0 = ± α q + 6 q 0 q c = ± α q + 6 q 0 q k = ± + l q + 6 q 0 q. 50 If q 0 = q = + m q = m φ this yields the exact solutions of Eq. as follows: 6 u x y t = ± α + m m sn + l + m x + ly t 6 u x y t = ± α + m ns + l + m x + ly t cnξ u 3 x y t = ± α + m + l + m x + ly t 5 6 u x y t = ± α + 6m + m m + ns ξ + l + 6 m + m x + ly t 5 3 u 5 x y t = ± α + 6m + m m nsξ + cnξ + l + 6m + m x + ly t 3 u 6 x y t = ± α 6m + m m + nsξ + cnξ + l 6m + m x + ly t. 53 If q 0 = m q = m q = φ this yields the exact solutions of Eq. as follows: 6 u 7 x y t = ± α m dn + l m x + ly t 6 m u 8 x y t = ± α m nd + l m x + ly t m cn ξ u 9 x y t = ± α m + l m x + ly t 5

7 60 M.M. Hassan Vol u 0 x y t = ± α m m ndξ m + l m + 6 x + ly t 55 m u x y t = ± α m + 6 m ndξ + i m cnξ m + l m + 6 x + ly t m u x y t = ± α m 6 + m ndξ + i m cnξ m + l m 6 x + ly t. 56 m With the aid of Appendix and choosing φ / + in formulas 50 we can obtain m u 3 x y t = ± α m + + l m x + ly t + u x y t = ± α m + l m x + ly t 6 u 5 x y t = ± α m cs ξ + l m x + ly t 57 m u 6 x y t = ± α + m csξ + l + m x + ly t m u 7 x y t = ± αm csξ + l m x + ly t. 58 Other JEFs are omitted here for simplicity. When m the solitary wave solutions of Eq. are obtained as follows: 3 u 8 x y t = ± α tanh + l x + ly t u 9 x y t = ± tanhξ + coth ξ α 8 + l x + ly t 59 6 u 0 x y t = ± α sech + lx + ly t u x y t = ± sechξ + i tanhξ α + l x + ly t 60 tanhξ u x y t = ± α + sechξ + l x + ly t + sechξ u 3 x y t = ± α tanhξ + l x + ly t 6 3 tanhξ u x y t = ± α + sechξ + sechξ tanhξ + cschξ + l x + ly t. 6 Remark Compared with the work in 7] where exact solutions for the S-KdV equation are studied we find more exact solutions in this paper. The solutions u x y t u x y t u 7 x y t u 8 x y t u 8 x y t u 0 x y t and u x y t coincide with the solutions given in 6] and other solutions of Eq. are new. Remark All the solutions obtained in this paper for Eq. 5 and Eq. have been checked by Maple software.

8 No. New Exact Solutions of Two Nonlinear Physical Models 603 Conclusion Applying the extended mapping method to the Schamel Korteweg-de Vries S-KdV equation and modified ZK equation we obtain a rich variety of new exact traveling wave solutions expressed by JEFs. These solutions include solitary wave solutions combined solitary wave solutions JEF solutions combined JEF solutions and rational function solutions. The solutions 3 37 and 38 are the same as the results obtained in 7] and other solutions to the best of our knowledge are new. Our new results ensure that the extended mapping method not only recovers all of the solutions reported in 7] but also discovers many new ones. When the modulus m new solitary wave solutions are also given. The present exact solutions may describe various new features of waves and may help one to find applications in plasma and dust plasma physics. Appendix: The ODE and Jacobi Elliptic Functions Table Relation between values of q 0 q q and corresponding φξ in ODE φ = q 0 + q φ + q φ. q 0 q q φ m m cd cnξ m m m cnξ m m m m ns dc cnξ m m m nc cnξ m m nd m m sc cnξ m m m sd m m cs cnξ m m m ds m m ± cnξ m ± m m m m m m m m + m + m + m + m m m ± icnξ m + i i m ± cnξ m ± cnξ m ± ± cnξ ± m cnξ ± mcnξ ± cnξ ± References ] M.J. Ablowitz and P.A. Clarkson Solitons Nonlinear Evolution Equations and Inverse Scattering Transform Cambridge University Press Cambridge 99. ] W. Malfliet Am. J. Phys ; A.H. Khater W. Malfliet D.K. Callebaut and E.S. Kamel Chaos Solitons & Fractals ] E.G. Fan Phys. Lett. A ; E.G. Fan and Y.C. Hong Z. Naturforsch 57a ] J.H. He and X.H. Wu Chaos Solitons & Fractals ] A.M. Wazwaz Comput. Math. Appl ; E. Yusufoglu A. Bekir and M. Alp Chaos Solitons & Fractals ] M. Wang Phys. Lett. A ; M. Wang Phys. Lett. A ] S.K. Liu Z.T. Fu S.D. Liu and Q. Zhao Phys. Lett. A ; Z.T. Fu S.K. Liu S.D. Liu and Q. Zhao

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