Computers and Mathematics with Applications. Stability analysis for Zakharov Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma

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1 Computers and Mathematics with Applications 67 () 7 8 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: Stability analysis for Zaharov Kuznetsov equation of wealy nonlinear ion-acoustic waves in a plasma A.R. Seadawy Mathematics Department, Faculty of Science and Arts, Taibah University, Al-Ula, Saudi Arabia Mathematics Department, Faculty of Science, Beni-Suef University, Egypt a r t i c l e i n f o a b s t r a c t Article history: Received 7 November Received in revised form July 3 Accepted November 3 Keywords: Traveling wave solutions Direct algebraic method Zaharov Kuznetsov equation Ion-acoustic waves The Zaharov Kuznetsov (ZK) equation is an isotropic nonlinear evolution equation, first derived for wealy nonlinear ion-acoustic waves in a strongly magnetized lossless plasma in two dimensions. In the present study, by applying the extended direct algebraic method, we found the electric field potential, electric field and magnetic field in the form of traveling wave solutions for the two-dimensional ZK equation. The solutions for the ZK equation are obtained precisely and the efficiency of the method can be demonstrated. The stability of these solutions and the movement role of the waves are analyzed by maing graphs of the exact solutions. 3 Elsevier Ltd. All rights reserved.. Introduction The Zaharov Kuznetsov (ZK) equation is a very attractive model equation for the study of vortices in geophysical flows. The ZK equation appears in many areas of physics, applied mathematics and engineering. In particular, it shows up in the area of Plasma Physics [ 3]. The ZK equation governs the behavior of wealy nonlinear ion-acoustic waves in a plasma comprised of cold ions and hot isothermal electrons in the presence of a uniform magnetic field [3 6]. The ZK equation [7] is one of two well-studied canonical two-dimensional extensions of the Korteweg de Vries equation [8]. In contrast to the Kadomtsev Petviashvilli (KP) equation, the ZK equation has so far never been derived in a geophysical fluid dynamics context. For a derivation of the KP equation for internal waves, see [9]. Traveling wave analysis is given in [] for the ZK equation. Soliton solutions are derived using the improved modified extended tanh-function method []. One-dimensional soliton, apparently inelastic [], periodic solutions [] and N-soliton solutions [3] have been obtained. The auxiliary equation method and the direct Hirota bilinear method were applied to the quantum ZK equation in [ 8]. This paper is organized as follows. In Section, the problem formulations to derive the nonlinear two-dimensional ZK equation are presented. In Section 3, the conservation laws for the ZK equation of wealy nonlinear ion-acoustic waves in a plasma are found. In Section, the Hamiltonian system for the momentum and the sufficient condition for soliton solution stability are given. In Section, the electric field potential, electric field and magnetic field in the form of traveling wave solutions of the ZK equation are obtained and analyzed. Finally the paper ends with a conclusion in Section 6.. Problem formulations Consider a low-β plasma in a magnetic field B = Be n, with T e T i, where T is the temperature and the subscripts i and e denote ions and electrons respectively. The ion motions are governed by n + (nv) =, t () Correspondence to: Mathematics Department, Faculty of Science, Beni-Suef University, Egypt. address: Aly7@yahoo.com. 898-/$ see front matter 3 Elsevier Ltd. All rights reserved.

2 v t + (v )v = e m i φ + v Ω i, φ = πe(n n e ), eφ n e = n o exp, T A.R. Seadawy / Computers and Mathematics with Applications 67 () where n is the number density of ions, v is their velocity, m i is the mass of an ion, φ is the electric field potential, and Ω i = eb. Let us non-dimensionalize the various quantities as follows: m i c n = n n, n e = n e n, mi v = v, Te x = x ρ, z = z L, t = t m i L T e, ρ = φ = φe, T e where L is the scale length of the waves. The independent variables are Te Ω i mi, () ζ = ϵ(z t), η = ϵx, τ = ϵ 3 t, (6) where ϵ is the small parameter characterizing the typical amplitude of the waves, and using the reduction perturbation method, from Eqs. () () the ZK equation can be derived as τ + µφ ζ + 3 φ ζ + 3 ( + δ) 3 φ η ζ where δ = λ D, λ = T e ρ D πn e. Consider the traveling wave solutions as =, (7) φ(ζ, η, τ) = φ(ξ), and ξ = ζ + lη + ωτ, (8) where, l and ω are wave numbers and frequency. Then Eq. (7) becomes ωφ + φφ + 3 φ (3) + l ( + δ)φ (3) =. (9) () (3) () 3. Conservation laws The ZK equation possesses some polynomial conservation laws, which can be cast into the following conservation forms: φ τ + φ + φ ζ ζ + ( + δ)φ ηη ζ =, () φ + φ 3 + 6φφ ζ ζ 3φζ + 6( + δ)φφ ηη + 3( + δ)φηζ ( + δ) φ ζ φ η =. () η τ Thus, if the electric field potential vanishes sufficiently rapidly at the ends of some intervals, it is easy to show that three integrals of motion (conserved quantities) exist for Eq. (7): I [φ] = φ dζ dη, () I [φ] = φ dζ dη, (3) I 3 [φ] = φ 3 + ζ + ( + δ) dζ dη. () η These integrals of motion were also found by Infeld [] using a variational formulation, and can provide rigorous information about the outcome of collisions of ZK solitary waves.. Stability analysis Eq. (7) is a Hamiltonian system for which the momentum is given by M = φ dξ ()

3 7 A.R. Seadawy / Computers and Mathematics with Applications 67 () 7 8 where M is the momentum and φ is the electric field potential. The sufficient condition for soliton stability is M ω > (6) where ω is the frequency.. Exact traveling wave solutions Now we shall find classes of solutions of the two-dimensional ZK equation, by applying the direct algebraic function method. The different values for φ give different analytic solutions of Eq. (7), which gives the following five cases. Case I. Assume that the ZK equation (9) has the following formal solution: m φ(ξ) = a i ϕ i (ξ), and (ϕ ) = αϕ + βϕ, (7) i= where α, β are arbitrary constants. Balancing the nonlinear term φφ and the highest order derivative φ (3) in Eq. (9) gives m =. The solution of Eq. (9) is in the form φ(ξ) = a + a ϕ + a ϕ. Substituting from (8) into Eq. (9) yields a set of algebraic equations for a, a, a,, l, ω, α, β, δ. The system of equations are found as 3 αa + l αa + l αδa + ωa + a a =, a + 3 αa + l αa + l αδa + ωa + a a =, 3 βa + l βa + l βδa a a =, 3 βa + l βa + l βδa a =. (9) The solution for the system of Eqs. (9), can be found as a = 3 α + l α + l αδ + ω, a =, a = 6β( + l + l δ). () Substituting from Eq. () into (8), the electric field potential of Eq. (7) can be obtained as a bell-shaped solitary wave solution φ(ζ, η, τ) = 3 α + l α + l αδ + ω The electric field E = φ = ζ e ζ η e η: (8) + 6βα( + l ( + δ)) sech α(ζ + lη + ωτ). () E = α 3/ ( + l ( + δ)) sech ( α(ζ + lη + ωτ)) tanh( α(ζ + lη + ωτ))(e ζ + le η ). () From Eq. (3), the difference between the number density of ions and electrons can be obtained as n n e = πe φ = 3 πe α ( + l )( + l ( + δ)) sech ( α(ζ + lη + ωτ))( + cosh( α(ζ + lη + ωτ))). (3) From the Maxwell Faraday equation E = B τ the magnetic field can be obtained in the form B = ω α3/ l( + l ( + δ)) sech ( α(ζ + lη + ωτ)) tanh( α(ζ + lη + ωτ))e n. () The electric field potential solution () is a Hamiltonian system for which the momentum is given by M = φ dξ. The sufficient condition for soliton solution stability is M ω >. () (6)

4 A.R. Seadawy / Computers and Mathematics with Applications 67 () a b Fig. (a-b). Electric field potential () with various different shapes: (a) bright solitary waves, (b) contour plot, in the interval [, ]. The bright solitary wave solution and contour plot for the electric field potential () are shown in Fig. (a) (b), the magnetic field () is represented by the solitary wave solution shown in Fig. (c) and the electric field is shown in Fig. (d) with =, l =., α =, β =., δ =.3, ω =, τ =., in the interval [, ]. According to the conditions of stability () (6), the electric field potential (), the electric field () and the magnetic field () are stable in the interval [, ]. Case II. Suppose that the solution of the ZK equation (7) has the following form: φ(ξ) = m a i ϕ i (ξ), and (ϕ ) = αϕ + βϕ 3 + γ ϕ, (7) i= where γ is arbitrary constant. Balancing the nonlinear term φφ and the highest order derivative φ (3) in Eq. (9) gives m =. The solution of Eq. (9) is of the form φ(ξ) = a + a ϕ + a ϕ. Substituting from (8) into Eq. (9) yields a set of algebraic equations for a, a, a,, l, ω, α, β, δ. The solution of the system of these algebraic equations can be found as a = α ( + l + l δ) ω, a = 3β( + l + l δ), a = 3β α ( + l + l δ), (8) γ = β α. (9) Substituting from Eq. (9) into (8), the electric field potentials can be obtained as the following exponential solutions of Eq. (7): φ (ζ, η, τ) = α ( + l + l δ) ω β( + l + l δ) exp(ζ + lη + ωτ) (exp(ζ + lη + ωτ) β) γ φ (ζ, η, τ) = ω γ ( + l + l δ) exp((ζ + lη + ωτ)) α (exp(ζ + lη + ωτ) β) γ, (3) β( + l + l δ) exp(ζ + lη + ωτ) β exp(ζ + lη + ωτ) + (β γ ) exp((ζ + lη + ωτ)) α ( + l + l δ) The electric fields E can be obtained as γ ( + l + l δ) exp((ζ + lη + ωτ)) α β exp(ζ + lη + ωτ) + (β γ ) exp((ζ + lη + ωτ)). (3) E = f (ζ, η, τ) f β + γ f β + β 3 f (β 8γ ) βγ ( + l ( + δ)) (f (ζ, η, τ) β) γ 3 (e ζ + le η ), (3) and E = f (β γ )f β + (β γ )βf (β 8γ )f ( + l ( + δ)) βf + (β γ )f 3 (e ζ + le η ), (33)

5 76 A.R. Seadawy / Computers and Mathematics with Applications 67 () 7 8 c d Fig. (c-d). The magnetic field and electric field: solitary wave solution in (c), vector plot in (d), in the intervals [, ] and [, ]. where f = f (ζ, η, τ) = exp(ζ + lη + ωτ). From the Maxwell Faraday equation, the magnetic field can be obtained in the form 3l( + l ( + δ)) β (β γ ) 3 f β(β γ ) (β γ ) βf (β + γ ) B = B = ω (f β) γ 3 f (β + γ β 9γ ) f 3 β(β + γ β 96γ ) + f (β γ )(β + 3γ ) + 3lβ f β ωγ γ (β γ )( + l ( + δ)) tanh e n (3) γ l( + l ( + δ)) f (β βf (β γ ) f (β 8γ ) + f 3 (β γ )(β 8γ )) ω f β + (β γ )f 3 lβ β ω γ ( + l ( + δ)) tanh (β γ )f e n. (3) γ The bright solitary wave solution and contour plot for the electric field potential (3) (3) are shown in Figs. (a) (d), with =., l =, α =., β = 3, δ =, ω =., τ =., γ =.; =, l =, α = 9, β = 3, δ =, ω =, τ = 3, γ = in the interval [, ] and [, ]. The electric field (3) is shown in Fig. (e) and the magnetic field (3) is represented solitary wave solution in Fig. (f) with =, l =, α =., β =., δ =, ω =.3, τ =., γ =, in the interval [, ] and [, ]. According to the conditions of stability () (6), the electric field potential (3) (3), the electric field (3) (33) and the magnetic field (3) (3) are stable in the interval [, ]; [, ]; [, ]. Case III. Assume that the solution of the ZK equation (9) is in the following form: φ(ξ) = a + a ϕ + a ϕ + a 3 ϕ 3 + a ϕ, (ϕ ) = αϕ + βϕ 6 (36) where a, a, a, a 3, a are arbitrary constants. Substituting from (36) into Eq. (9) yields a set of algebraic equations for a, a, a, a 3, a,, l, ω, α, β, δ, γ. The solution of the system of these algebraic equations can be found as a = ω, a =, a = 6α( + l + l δ), a 3 =, a = β( + l + l δ). (37) Substituting from Eq. (37) into (36), the following solution of Eq. (7) can be obtained for the electric field potential in the form of rational solution as φ(ζ, η, τ) = ω 6α ( + l + l δ) α (ζ + lη + ωτ) β ± α β( + l + l δ) (α (ζ + lη + ωτ) β). (38) The electric fields E can be obtained as E = α3 ( + l ( + δ))(ζ + lη + ωτ) βα + α(8β + α (ζ + lη + ωτ) ) α (ζ + lη + ωτ) β 3 (e ζ + le η ). (39)

6 A.R. Seadawy / Computers and Mathematics with Applications 67 () b a 6 Fig. (a-b). Electric field potential (3): (a) bright solitary waves, (b) contour plot, in the interval [, ]. c d...6 Fig. (c-d). Electric field potential (3): (c) bright solitary waves, (d) contour plot, in the intervals [, ] and [, ]. e f Fig. (e-f). The electric field and magnetic field: vector plot in (e), solitary wave solution in (f), in the intervals [, ] and [, ]. From the Maxwell Faraday equation, the magnetic field can be obtained in the form = B 6α 3 ℓ( + ℓ ( + δ)) ξ ( αγ + γ α ξ + αγ (8β + α ξ )) 3 γ ω αξ γ 6α 3 ℓ( + ℓ ( + δ)) αξ tanh en. ωγ γ γ () The bright solitary wave solution and contour plot for the electric field potential (38) are shown in Fig. 3(a) (b), with =., ℓ =., α =, β =., δ =., ω =, τ =, γ = in the interval [, ] and [, ].

7 78 A.R. Seadawy / Computers and Mathematics with Applications 67 () 7 8 a b... Fig. 3. The electric field potential (38) are plotted: solitary wave solution in 3(a), contour plot in 3(b), in the intervals [, ] and [, ]. Case IV. Suppose that the ZK equation (9) has the following formal solution: φ(ξ) = a + a ϕ + a ϕ, and (ϕ ) = αϕ + βϕ + γ ϕ 6. () Substituting from () into Eq. (9) yields a set of algebraic equations for a, a, a,, l, ω, α, β, γ, δ. The solution of the system of these algebraic equations can be found as a = α3 + αl ( + δ) + ω, a =, a = 6β( + l ( + δ)). () Substituting from Eq. () into (), the solution of Eq. (7) can be obtained as φ (ζ, η, τ) = α3 + αl ( + δ) + ω φ (ζ, η, τ) = α3 + αl ( + δ) + ω The electric fields E can be obtained as β(α + αl (ζ +lη+ωτ) ( + δ))e, β γ βe (ζ +lη+ωτ) (3) + e(ζ +lη+ωτ) β( + l (ζ +lη+ωτ) ( + δ))e. βe (ζ +lη+ωτ) () + (β γ )e(ζ +lη+ωτ) E = 8βe(ζ +lη+ωτ) β γ + e (ζ +lη+ωτ) ( + l ( + δ)) β + γ e (ζ +lη+ωτ) (e ζ + le η ). () E = 8βe(ζ +lη+ωτ) (γ β)e (ζ +lη+ωτ) ( + l ( + δ)) γ e (ζ +lη+ωτ) + γ β e (ζ +lη+ωτ) (e ζ + le η ). (6) From the Maxwell Faraday equation, the magnetic field can be obtained in the form B = 8βe(ζ +lη+ωτ) (γ β) e (ζ +lη+ωτ) ( l )( + l ( + δ)) ω γ β γ e (ζ +lη+ωτ) + e (ζ +lη+ωτ) e n. (7) Figs. (a) (d), represent the traveling wave solutions (3) () of the ZK equation (7) and contour plot, with =., l =., β =, δ =.8, ω =, τ =., γ =, α = and =., n =., β =, δ =, ω =, τ =., γ = 3 in the intervals [, ] and [, ]. Case V. The solution of the ZK equation (9) has the following form: φ(ξ) = a + a ϕ + a ϕ, ϕ = α + βϕ + γ ϕ. (8) Substituting from (8) into Eq. (9) yields a set of algebraic equations for a, a, a,, l, ω, α, β, γ, δ. The solution of the system of these algebraic equations can be found as a = α(α + 8β)( + l ( + δ)) ω, a = 6αβ( + l ( + δ)), a = 6β ( + l ( + δ)). (9)

8 A.R. Seadawy / Computers and Mathematics with Applications 67 () a b 6 7 Fig. (a-b). The electric field potential (3): solitary wave solution in 3(a), contour plot in 3(b), in the intervals [, ] and [, ]. c d Fig. (c-d). The electric field potential (): solitary wave solution in (c), contour plot in (d), in the intervals [, ] and [, ]. Substituting from Eq. (9) into (8), the solution of Eq. (7) can be obtained in the form of the Riccati sub-equation method as φ(ζ, η, τ) = α(α + 8β)( + l ( + δ)) ω β + 6αβ( + l ( + δ)) γ 6β ( + l ( + δ)) β γ + The electric fields E can be obtained as E = 3β γ (αγ β ) + l ( + δ) sec γ α β + β αγ β tan αγ β γ αγ β γ tan αγ β /(ζ + lη + ωτ) tan αγ β /(ζ + lη + ωτ). () αγ β (ζ + lη + ωτ) αγ β (ζ + lη + ωτ) (e ζ + le η ). () From the Maxwell Faraday equation, the magnetic field can be obtained in the form B = lβ γ ω (αγ β ) + l ( + δ) tan αγ β (ζ + lη + ωτ) 3γ α 3β + β αγ β tan αγ β (ζ + lη + ωτ) e n. () Both dar solitary wave solutions and contour plot () are shown in Fig. (a) (b), with =., l =., β =, δ = 3, ω =, τ =, γ = in the intervals [, ] and [ 3, 3].

9 8 A.R. Seadawy / Computers and Mathematics with Applications 67 () 7 8 a b Fig.. The electric field potential () are plotted: solitary wave solution in (a), contour plot in (b), in the intervals [, ] and [ 3, 3]. 6. Conclusion The problem formulations of wealy nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field is given for the nonlinear two-dimensional ZK equation. By implementing the extended direct algebraic method, we present traveling wave solutions for the nonlinear two-dimensional ZK equation. The stability analysis for the electric field potentials, electric fields and magnetic fields are discussed with respect to the sufficient condition for soliton stability. We obtained in case I the electric field potential in the form of a bell-shaped solitary wave solution. The electric field potential, electric field and magnetic field are stable in the interval [, ]. In cases II and IV, we obtained solutions in the form of exponential solitary wave solutions and these solutions are different in form and coefficients from the Ji-Huan He solutions. The solutions in form rational solitary wave solutions are given in case III. In case V, we obtained the electric field potential in the form of dar solitary wave solution which is stable in the interval [, ]. Acnowledgment This project was supported by the Deanship of Scientific Research, Taibah University, KSA, project No. 79, year: 3. References [] J. Das, A. Bandyopadhyay, K.P. Das, Existence and stability of alternatives ion-acoustic solitary wave solution of the combined MKdV KdV ZK equation in a magnetized non thermal plasma consisting of warm adiabatic ions, Phys. Plasmas (9) (7) 93. [] C. Lin, X. Zhang, The formally variable separation approach for the modified Zaharov Kuznetsov equation, Commun. Nonlinear Sci. Numer. Simul. () (7) [3] A. Mushtaq, H.A. Shah, Nonlinear Zaharov Kuznetsov equation for obliquely propagating two-dimensional ion-acoustic waves in a relativistic, rotating magnetized electron positron ion plasma, Phys. Plasmas (7) () 736. [] X. Lu, B. Tian, T. Xu, K.-J. Cai, W.-J. Liu, Analytical study of the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential in quasi-one-dimensional Bose Einstein condensates, Ann. Physics 33 () (8) 6. [] X. Lu, H.-W. Zhu, Z.-Z. Yao, X.-H. Meng, C. Zhang, C.-Y. Zhang, B. Tian, Multisoliton solutions in terms of double Wronsian determinant for a generalized variable coefficient nonlinear Schrödinger equation from plasma physics, arterial mechanics, fluid dynamics and optical communications, Ann. Physics 33 (8) (8) [6] A. Biswas, E. Zerrad, Solitary wave solution of the Zaharov Kuznetsov equation in plasmas with power law nonlinearity, Nonlinear Anal. RWA () [7] V.E. Zaharov, E.A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP 39 (97) [8] B.B. Kadomtsev, V.I. Petviashvilli, On the stability of solitary waves in wealy dispersing media, Sov. Phys. Dol. (97) 3. [9] R.H.J. Grimshaw, Y. Zhu, Oblique interaction between internal solitary waves, Stud. Appl. Math. 9 (99) 9. [] I. Kourais, W.M. Moslem, U.M. Abdelsalam, R. Sabry, P.K. Shula, Nonlinear dynamics of rotating multi-component pair plasmas and epi plasmas, Plasma Fusion Res. (9). [] E. Infeld, P. Fryczs, Self-focusing of nonlinear ion-acoustic waves and solitons in magnetized plasmas. Part. Numerical simulations, two-soliton collisions, J. Plasma Phys. 37 (987) [] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals 6 () [3] Q. Qu, B. Tian, W. Liu, K. Sun, P. Wang, Y. Jiang, B. Qin, Soliton solutions and interactions of the Zaharov Kuznetsov equation in the electron positron ion plasmas, Eur. Phys. J. D 6 () [] B. Zhang, Z. Liu, Q. Xiao, New exact solitary wave and multiple soliton solutions of quantum Zaharov Kuznetsov equation, Appl. Math. Comput. 7 (). [] F. Awawdeh, New exact solitary wave solutions of the Zaharov Kuznetsov equation in the electron positron ion plasmas, Appl. Math. Comput. 8 () [6] A.R. Seadawy, New exact solutions for the KdV equation with higher order nonlinearity by using the variational method, Comput. Appl. Math. 6 () [7] A.R. Seadawy, Traveling wave solutions of the Boussinesq and generalized fifth-order KdV equations by using the direct algebraic method, Appl. Math. Sci. 6 () 8 9. [8] A.R. Seadawy, K. El-Rashidy, Traveling wave solutions for some coupled nonlinear evolution equations by using the direct algebraic method, Math. Comput. Modelling (3).