The extended Jacobi Elliptic Functions expansion method and new exact solutions for the Zakharov equations

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1 ISSN England UK World Journal of Modelling and Simulation Vol. 5 (2009) No. 3 pp The extended Jacobi Elliptic Functions expansion method and new exact solutions for the Zakharov equations Baojian Hong Dianchen Lu 2 Fushu Sun Department of Basic Courses Nanjing Institute of Technology Nanjing 267 P. R. China 2 Faculty of Science Jiangsu University Zhenjiang 2203 P. R. China (Received July Revised April Accepted June ) Abstract. We extended the Jacobi elliptic function expansion method by constructing four new Jacobian elliptic functions and apply this method to Zakharov equations for illustration abundant new doubly periodic solutions are obtained these solutions are degenerated to the solitary wave solutions and the triangle function solutions in the limit cases when the modulus of the Jacobian elliptic functions m or 0 which shows that the new method is more powerful to seek the exact solutions of the nonlinear partial differential equations in mathematical physics. Keywords: Zakharov equations extended Jacobian Elliptic Functions expansion method doubly periodic solutions exact solutions Introduction In the interaction of laser-plasma the system of Zakharov equation (SZE) plays an important role [8 9]. This system attracted many scientists wide interest and attention. In one dimension the formation evolution and interaction of the Langmuir solution differ from solutions of the KdV equation. In multi-dimensions the Langmuir solution will collapse. Since 980s the effects including magnetic field have been considered and the system of Zakharov equation includes more general form and rich contents. For example SZE with Landau damping effect was given in [8 4]. Under some conditions its inverse scattering transformation has been found. In [8] the longitudinal and transverse oscillating and magnetic field effect was examined the solution properties and collapse in multi-dimensions have been revealed. More recently some authors considered the exact and explicit solutions of the system of Zakharov equations by different methods in [ ]. In this paper we consider the system of Zakharov equations by constructing four new types of Jacobian elliptic functions and abundant new families of exact solutions are obtained. This paper is arranged as follows. In section 2 we briefly describe the new extended Jacobi elliptic functions expansion method. In section 3 several families of solutions to the Zakharov equations are obtained which are degenerated to new solitary wave solutions and new periodic wave solutions in the limit case. In section 4 some conclusions are given. 2 Summary of the extended jacobi elliptic functions expansion method In this section we propose a general method namely the extended Jacobi elliptic functions expansion method given nonlinear partial differential equation for instance in two variables x and t as follows: This research was Supported by Scientific Research Foundation of NanJing Institute of Technology (Grant No. KXJ08047). Corresponding author. address: hongbaojian@63.com. Published by World Academic Press World Academic Union

2 World Journal of Modelling and Simulation Vol. 5 (2009) No. 3 pp P (u u t u x u xx ) = 0 () We seek the following formal solutions of the given system by a new intermediate transformation: u(ξ) = n a i F i (ξ) + i= n b i F i (ξ)e(ξ) + i= n c i F i (ξ)g(ξ) + i= n d i F i (ξ)h(ξ) + a 0 (2) where a 0 a i b i c i d i (i = 2 n) are constants to be determined later. ξ = ξ(x t) are arbitrary functions with the variables x and t The parameter n can be determined by balancing the highest order derivative terms with the nonlinear terms in Eq. (). And E(ξ) F (ξ) G(ξ) H(ξ) are an arbitrary array of the four function e = e(ξ) f = f(ξ) g = g(ξ) and h = h(ξ) the selection obey the principle which makes the calculation more simple. Here we get e = p + qsn[ξ m] + rcn[ξ m] + ldn[ξ m] f = sn[ξ m] p + qsn[ξ m] + rcn[ξ m] + ldn[ξ m] cn[ξ m] g = p + qsn[ξ m] + rcn[ξ m] + ldn[ξ m] h = dn[ξ m] p + qsn[ξ m] + rcn[ξ m] + ldn[ξ m] where p q r l are arbitrary constants the four function e f g h satisfy the following restricted relation { e = qgh + rfh + lm 2 fg f = pgh + reh + leg g = pfh qeh + l(m 2 )ef h = m 2 pfg r(m 2 )ef qeg g 2 = e 2 f 2 h 2 = e 2 m 2 f 2 (4) where denotes dξ d. m is the modulus of the Jacobi elliptic function (0 m ) and e f g h satisfy the following six relation: Family : when l = p = 0 we have qf + rg =. (5) Family 2 : when l = r = 0 we have pe + qf =. (6) Family 3 : when l = q = 0 we have pe + rg =. (7) Family 4 : when p = r = 0 we have lh + qf =. (8) Family 5 : when p = q = 0 we have lh + rg =. (9) Family 6 : when q = r = 0 we have lh + pe =. (0) Here we can select F(ξ) = g(ξ) in Eq. (7) and Eq. (9) F(ξ) = f(ξ) in Eq. (5) Eq. (6) Eq. (8) and F (ξ) = e(ξ) in Eq. (0) Substituting Eq. (2) and Eq. (4) along with Eq. (5) Eq. (0) into Eq. () separately yields six families of polynomial equations for E(ξ) F (ξ) G(ξ) H(ξ). Setting the coefficients of F i (ξ) F i (ξ)e(ξ) F i (ξ)g(ξ) F i (ξ)h(ξ) F i (ξ)e(ξ)g(ξ) F i (ξ)e(ξ)h(ξ) F i (ξ)g(ξ)h(ξ) (i = 0 2 ) to zero yields a set of nonlinear algebraic equations (NAEs) in a 0 a i b i c i d i (i = 2 n) and ξ(x t) solving the NAEs by Mathematica and Wu elimination we can obtain many exact solutions of Eq. () according to Eq. (2) Eq. (4) and Eq. (5) Eq. (0). Obviously if we choose the special value of p q r l in Eq. (3) then we can get the result of [4-2]. For example when we choose p = in Eq. (6) Eq. (7) and Eq. (0) then we can get all the results in [4 5] when we choose m r = in Eq. (5) and Eq. (7) we can get the results in [7-9] when we choose three of them to be zero we can get the results in [6 20-2]. Remark. The restricted equations Eq. (4) Eq. (5) Eq. (0) and the solutions of Eq. (3) are new they contain the results in [7] [6] completely while the form turns much simplified. Their practical value turns stronger. Remark 2. Here the value of i can be extended to i = n n so the method contain plenty of Jacobi function expansion methods [ ] noticed that snξ tanh ξ cnξ sec hξ dnξ sec hξ when the modulus m and snξ sin ξ cnξ cos ξ dnξ when the modulus m 0 we can obtain the corresponding solitary wave solutions and triangle function solutions. In the following we will use this method to solve the zakharov equations. i= (3) WJMS for subscription: info@wjms.org.uk

3 28 B. Hong & D. Lu & et al.: The extended Jacobi elliptic functions expansion method 3 Exact solutions to the zakharov equations We consider the following system of Zakharov Eq. () Eq. (4). { u tt c 2 su xx β( v 2 ) xx = 0 iv t + αv xx uv = 0 (a) (b) () where u = u(x t) is the perturbed number density of the ion (in the low-frequency response) v = v(x t) is the slow variation amplitude of the electric field intensity c s is the thermal transportation velocity of the electron-ion α 0 β 0 0 c s are constants. Eq. (a) and Eq. (b) are one of the fundamental models governing dynamics of nonlinear waves and describe the interactions between high- and low- frequency waves. The physically most important example involves the interaction between Langmuir and ion-acoustic waves in plasma. Since v(x t) is a complex function thus we introduce a gauge transformation: u = u(x t) = u(ξ) v = v(x t) = v(ξ) = φ(ξ) exp[i(sx ωt)] ξ = k(x ct) + ξ 0 (2a) (2b) (2c) where φ(ξ) is a real-valued function s ω k c are four real constants to be determined and ξ 0 is an arbitrary constant. Substituting (2) into () we have: { k 2 (c u βk 2 (φ 2 ) = 0 (3a) αk 2 φ + (ω αs 2 (3) )φ uφ + i(2αsk kc)φ = 0 (3b) Integrating Eq. (3a) twice with respect to u and put the integration constants to zero we obtain (2) u = β c 2 c 2 φ 2 (4) s Let c = 2αs and substituting (4) into (3b) we obtain: αk 2 φ + (ω αs 2 )φ β c 2 c 2 φ 3 = 0 (5) s By the homogenous balance principle we have n= thus we assume that the Liènard Eq. (5) have the following solutions: φ = c 0 + c e + c 2 f + c 3 g + c 4 h (6) where φ = φ(ξ) e = e(ξ) f = f(ξ) g = g(ξ) h = h(ξ) and e f g h satisfy ((4) (0)). Substituting (4) and ((5) (0)) along with (2) and (6) into (5) and setting the coefficients of F i (ξ) F i (ξ)e(ξ) F i (ξ)g(ξ) F i (ξ)h(ξ) F i (ξ)e(ξ)g(ξ) F i (ξ)e(ξ)h(ξ) F i (ξ)g(ξ)h(ξ) (i = 0 2 ) to zero yields nonlinear algebraic equations (NAEs) with respect to the unknown k ω s c c i (i = ) p q r l. We could determine the following solutions of Zakharov Eq. (). Family Case. ω = (k 2 ( 2m 2 ) + s 2 )α c 0 = c 2 = c 3 = c 4 = 0 c = ±kr Case 2. ω = (k 2 ( m2 2 ) + s2 )α c 0 = c 2 = c 3 = c 4 = 0 c = ±kmr 2α( m 2 )(c 2 c 2 s) 2 s c 2 ) 2 q = εir WJMS for contribution: submit@wjms.org.uk

4 World Journal of Modelling and Simulation Vol. 5 (2009) No. 3 pp Case 3. ω = (k 2 ( + m 2 ) + s 2 )α c 0 = c = c 2 = c 3 = 0 c 4 = ±kr 2 2 c 2 s) Case 4. ω = (k 2 ( m2 2 ) + s2 )α c 0 = c = c 2 = c 4 = 0 c 3 = ±krm 2 c 2 s) 2 q = εir m 2 ω = (k 2 ( m2 2 ) + α(q s2 )α c 0 = c 2 = c 3 = 0 c = ±k 2 + ( m 2 )r 2 )(c 2 Case 5. α(q c 4 = εk 2 + r 2 )(c 2 where c = 2αs ε = ± i = p q r l k s are arbitrary constants so do the following situations. Therefore from Eq. (3) Eq. (5) Eq. (0) Eq. (2) Eq. (4) Eq. (6) and Cases 5 we obtain the solutions to the Eq. (): u = 2αk2 ( m 2 ) nc 2 [ξ m] 2α( m v = ±k 2 )(c nc[ξ m] exp[i(sx (k 2 ( 2m 2 ) + s 2 )αt)] u 2 = αk2 m 2 2 (εisn[ξ 2 m] + cn[ξ 2 m]) 2 v 2 = ±km 2 s c 2 ) exp[i(sx (k 2 ( m2 2 ) + s2 )αt)] 2 εisn[ξ 2 m] + cn[ξ 2 m] u 3 = 2αk2 dc 2 [ξ 3 m] 2 v 3 = ±k dc[ξ 3 m] exp[i(sx (k 2 ( + m 2 ) + s 2 )αt)] u 4 = αk2 m 2 cn 2 [ξ 4 m] 2 (εi m 2 sn[ξ 4 m] + cn[ξ 4 m]) 2 v 4 = ±km 2 c 2 cn[ξ 4 m] exp[i(sx (k s) 2 ( m2 2 ) + s2 )αt)] 2 εi m 2 sn[ξ 4 m] + cn[ξ 4 m] u 5 = αk2 2 v 5 = k (± q 2 + ( m 2 )r 2 + ε q 2 + r 2 dn[ξ 5 m]) 2 (qsn[ξ 5 m] + rcn[ξ 5 m]) 2 2 exp[i(sx (k 2 ( m2 2 ) + s2 )αt)] where ξ i = k(x 2αst) + ξ 0 (i = 5). (± q 2 + ( m 2 )r 2 + ε q 2 + r 2 dn[ξ 5 m]) qsn[ξ 5 m] + rcn[ξ 5 m] Family 2 Case 6. ω = (k 2 ( 2m 2 ) + s 2 )α c 0 = c = c 2 = c 4 = 0 c 3 = ±kmp 2 2 c 2 s) Case 7. ω = 2 (k2 ( + m 2 ) 2s 2 )α c 0 = c = c 2 = c 4 = 0 c 3 = ±kp α( m 2 )(c 2 c 2 s) 2 q = εp WJMS for subscription: info@wjms.org.uk

5 220 B. Hong & D. Lu & et al.: The extended Jacobi elliptic functions expansion method ω = 2 (k2 ( + m 2 ) 2s 2 α(q )α c 0 = c = c 2 = 0 c 3 = ±k 2 m 2 p 2 )(c 2 Case 8. α(q c 4 = εk 2 p 2 )(c 2 Case 9. ω = (k 2 ( + m 2 ) + s 2 )α c 0 = c = c 3 = c 4 = 0 c 2 = ±kmp 2 2 c 2 s) We can derive the following solutions: u 6 = 2αβk2 m 2 cn 2 [ξ 6 m] 2 v 6 = ±km cn[ξ 6 m] exp[i(sx (k 2 ( 2m 2 ) + s 2 )αt)] u 7 = αk2 ( m 2 ) cn 2 [ξ 7 m] 2 ( + εsn[ξ 7 m]) 2 α( m v 7 = ±k 2 )(c 2 c 2 cn[ξ 7 m] exp[i(sx + s) 2 (k2 ( + m 2 ) 2s 2 )αt)] 2 + εsn[ξ 7 m] u 8 = αk2 2 v 8 =k (±k q 2 m 2 p 2 cn[ξ 8 m] + ε q 2 p 2 dn[ξ 8 m]) 2 (p + qsn[ξ 8 m]) 2 2 c 2 (±k q s) 2 m 2 p 2 cn[ξ 8 m]+ε q 2 p 2 dn[ξ 8 m]) exp[i(sx+ 2 (k2 (+m 2 ) 2s 2 )αt)] 2 p+qsn[ξ 8 m] u 9 = 2αk2 m 2 sn 2 [ξ 9 m] 2 v 9 = ±km sn[ξ 9 m] exp[i(sx (k 2 ( + m 2 ) + s 2 )αt)] where ξ i = k(x 2αst) + ξ 0 (i = 6 9). Family 3 Case 0. ω = (k 2 ( 2 + m2 ) + s 2 )α c 0 = c = c 3 = 0 c 2 = ±k α(p c 4 = εk 2 r 2 )(c 2 Case. ω = (k 2 (m 2 2 ) + s2 )α c 0 = c = c 3 = c 4 = 0 c 2 = ±kp Case 2. ω = (k 2 (m 2 2 ) + s2 )α c 0 = c = c 2 = c 3 = 0 c 4 = ±kp Case 3. ω = 2 (k2 ( + m 2 ) 2s 2 )α c 0 = c = c 2 = 0 c 3 = ±kmp c 4 = εkp r = 0 2 We derive the following solutions of Eq. (). α(r 2 + m 2 (p 2 r 2 ))(c 2 r = εp 2 2(m 2 ) r = 2 εmp m 2 WJMS for contribution: submit@wjms.org.uk

6 World Journal of Modelling and Simulation Vol. 5 (2009) No. 3 pp u 0 = αk2 (± r 2 + m 2 (p 2 r 2 )sn[ξ 0 m] + ε r 2 p 2 )dn[ξ 0 m]) 2 2 (p + rcn[ξ 0 m]) 2 v 0 = k (± r 2 + m 2 (p 2 r 2 )sn[ξ 0 m] + ε r 2 p 2 )dn[ξ 0 m]) 2 p + rcn[ξ 0 m] exp[i(sx (k 2 ( 2 + m2 ) + s 2 )αt)] u = αk2 sn 2 [ξ m] 2 ( + εcn[ξ m]) 2 v = ±k sn[ξ m] 2 + εcn[ξ m] exp[i(sx (k2 (m 2 2 ) + s2 )αt)] u 2 = αk2 dn 2 [ξ 2 m] 2 ( m 2 + εmcn[ξ 2 m]) 2 v 2 = ±k dn[ξ 2 m] 2 m 2 + εmcn[ξ 2 m] exp[i(sx (k2 (m 2 2 ) + s2 )αt)] u 3 = αk2 2 (±mcn[ξ 3 m] + εdn[ξ 3 m]) 2 v 3 = k (±mcn[ξ 2 3 m] + εdn[ξ 3 m]) exp[i(sx + 2 (k2 ( + m 2 ) 2s 2 )αt)] where ξ i = k(x 2αst) + ξ 0 (i = 0 3). Family 4 Case 4. ω = (k 2 (m 2 2) + s 2 )α c 0 = c 2 = c 3 = c 4 = 0 c = ±kl 2α(m 2 )(c 2 c 2 s ) Case 5. ω = (k 2 (m 2 2 ) + s2 )α c 0 = c 2 = c 3 = c 4 = 0 c = ±kl 2 c 2 s ) 2 q = εiml Case 6. ω = (k 2 (m 2 2 ) + s2 )α c 0 = c = c 2 = c 4 = 0 c 3 = ±kl 2 c 2 s ) 2 q = εl m 2 Case 7. ω = (k 2 ( 2 m2 ) + s 2 )α c 0 = c = c 3 = 0 c 2 = ±kml α(m 2 )(c 2 c 2 s) c 3 = εkml 2 c 2 s) 2 2 We derive the following solutions of Eq. (). u 4 = 2αk2 (m 2 ) nd 2 [ξ 4 m] 2α(m v 4 = ±k 2 )(c nd[ξ 4 m] exp[i(sx (k 2 (m 2 2) + s 2 )αt)] u 5 = αk2 2 v 5 = ±k (εimsn[ξ 5 m] + dn[ξ 5 m]) 2 2 εimsn[ξ 5 m] + dn[ξ 5 m] exp[i(sx (k2 (m 2 2 ) + s2 )αt)] WJMS for subscription: info@wjms.org.uk

7 222 B. Hong & D. Lu & et al.: The extended Jacobi elliptic functions expansion method u 6 = β αk c 2 c 2 ± k 2 (c cn 2 [ξ 6 m] 2 s (ε m 2 sn[ξ 6 m] + dn[ξ 6 m]) 2 v 6 = ±k 2 c 2 cn[ξ 6 m] exp[i(sx (k s) 2 (m 2 2 ) + s2 )αt)] 2 ε m 2 sn[ξ 6 m] + dn[ξ 6 m] u 7 = αk2 m 2 (± m 2 2 sd[ξ 7 m] + εcd[ξ 7 m]) 2 v 7 = km (± m 2 2 sd[ξ 7 m] + εcd[ξ 7 m]) exp[i(sx (k 2 ( 2 m2 ) + s 2 )αt)] where ξ i = k(x 2αst) + ξ 0 (i = 4 7). Family 5 Case 8. ω = 2 (k2 (m 2 + ) 2s 2 )α c 0 = c 2 = c 3 = c 4 = 0 c = ±kl( m 2 ) 2 c 2 s ) 2 r = εml Case 9. ω = (k 2 ( 2m 2 ) + s 2 )α c 0 = c = c 3 = c 4 = 0 c 2 = ±kml 2α(m 2 )(c 2 c 2 s) r = 0 Case 20. ω = 2 (k2 (m 2 + ) 2s 2 )α c 0 = c = c 3 = c 4 = 0 c 2 = ±kl( m 2 ) We derive the following solutions of Eq. (). u 8 = αk2 ( m 2 ) 2 2 v 8 = ±k( m 2 ) (εmcn[ξ 8 m] + dn[ξ 8 m]) 2 2 c 2 exp[i(sx + s) 2 (k2 (m 2 + ) 2s 2 )αt)] 2 εmcn[ξ 8 m] + dn[ξ 8 m] 2 c 2 s) 2 r = εl u 9 = 2αk2 m 2 (m 2 ) sd 2 [ξ 9 m] v 9 = ±km 2α(m 2 )(c sd[ξ 9 m] exp[i(sx (k 2 ( 2m 2 ) + s 2 )αt)] u 20 = αk2 ( m 2 ) 2 sn 2 [ξ 20 m] 2 (εcn[ξ 20 m] + dn[ξ 20 m]) 2 v 20 = ±k( m 2 ) 2 c 2 sn[ξ 20 m] exp[i(sx + s) 2 (k2 (m 2 + ) 2s 2 )αt)] 2 εcn[ξ 20 m] + dn[ξ 20 m] where ξ i = k(x 2αst) + ξ 0 (i = 8 20). Family 6 Case 2. ω = (k 2 (m 2 2) + s 2 )α c 0 = c = c 2 = c 3 = 0 c 4 = ±kp 2 2 c 2 s ) l = 0 Case 22. ω = (k 2 ( 2 m2 ) + s 2 )α c 0 = c = c 2 = c 4 = 0 c 3 = ±km 2 p 2 c 2 s) 2( m 2 ) l = Case 23. ω = (k 2 ( 2 m2 ) + s 2 )α c 0 = c = c 3 = c 4 = 0 c 2 = ±km 2 p 2 c 2 s) 2 l = εp εp m 2 WJMS for contribution: submit@wjms.org.uk

8 World Journal of Modelling and Simulation Vol. 5 (2009) No. 3 pp ω = (k 2 ( 2 m2 ) + s 2 α(p )α c 0 = c = c 4 = 0 c 2 = km 2 + (m 2 )l 2 )(c 2 Case 24. α(p c 3 = εkm 2 l 2 )(c 2 We derive the following solutions of Eq. (). u 2 = 2αk2 dn 2 [ξ 2 m] 2 v 2 = ±k dn[ξ 2 m] exp[i(sx (k 2 (m 2 2) + s 2 )αt)] u 22 = αk2 m 4 2 v 22 = ±km 2 dn 2 [ξ 22 m] ( m 2 + εdn[ξ 22 m]) 2 2 c 2 dn[ξ 22 m] exp[i(sx (k s) 2 ( 2 m2 ) + s 2 )αt)] 2 m 2 + εdn[ξ 22 m] u 23 = αk2 m 4 2 v 23 = ±km 2 sn 2 [ξ 23 m] ( + εdn[ξ 23 m]) 2 2 sn[ξ 23 m] + εdn[ξ 23 m] exp[i(sx (k2 ( 2 m2 ) + s 2 )αt)] u 24 = αk2 m 2 2 v 24 = km [ exp i ( p 2 + (m 2 )l 2 sn[ξ 24 m] + ε l 2 p 2 cn[ξ 24 m]) 2 (p + ldn[ξ 24 m]) 2 2 ( sx (k 2 ( 2 m2 ) + s 2 )αt ( p 2 + (m 2 )l 2 sn[ξ 24 m] + ε l 2 p 2 cn[ξ 24 m]) p + ldn[ξ 24 m] )] where ξ i = k(x 2αst) + ξ 0 (i = 2 24). Remark 3. Here u 7 v 7 ; u v ; u 2 v 2 ; u 23 v 23 contain the result of (3) (27); (8) (32); (9); (23) (37) in Ref. [6] if we let q = ±mp; q = r p = or p = r q = in u 8 v 8 we can get the result of (4)(5)(29) in Ref. [6] if we let p = r = or r = p = r in u 0 v 0 we can get the result of (32) (34) in Ref. [6] if we let p = l = r or p = r l = in u 24 v 24 we can get the result of (25) (39) in Ref. [6]. u v ; u 4 v 4 ; u 9 v 9 contain the result of u 2 v 2 ; u 8 v 8 ; u 9 v 9 ; in Ref. [7] if we let l = 0 in u 24 v 24 and r = 0 in u 0 v 0 we can get the result of (5-8) in Ref. [7]. Remark 4. It is notable that the other types of solutions we obtained here to systems () are not shown in the previous literature. They are degenerated to the corresponding solitary wave solutions and triangle function solutions in the limit cases when m or m 0. 4 Conclusion In this paper we succeed to propose an approach for finding new exact solutions for nonlinear evolution equations by constructing the four new types of Jacobian elliptic functions. By using this method and computerized symbolic computation we have found thirteen new types of exact solutions for the zakharov Eq. (). More importantly our method is much simple and powerful to find new solutions to various kinds of nonlinear evolution equations such as KdV equation mkdv equation Boussinesq equation etc. We believe that this method should play an important role for finding exact solutions in the mathematical physics. WJMS for subscription: info@wjms.org.uk

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