IMA Preprint Series # 2014

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1 GENERAL PROJECTIVE RICCATI EQUATIONS METHOD AND EXACT SOLUTIONS FOR A CLASS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS By Emmanuel Yomba IMA Preprint Series # 2014 ( December 2004 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY OF MINNESOTA 514 Vincent Hall 206 Church Street S.E. Minneapolis, Minnesota Phone: 612/ Fax: 612/ URL:

2 General Projective Riccati equations method and exact solutions for a class of nonlinear partial differential equations Emmanuel Yomba a,b a Institute for Mathematics and its applications, University of Minnesota, 400 Lind Hall 207 Church Street S.E. Minneapolis, MN U.S.A. b Department of Physics, Faculty of Sciences, University of Ngaoundéré PO. BOX 454 Ngaoundéré Cameroon. ABSTRACT By using a simple transformation technique, we have shown that the Hamiltonian amplitude equation, the nonlinear wave equation, the coupled Klein-Gordon-Zakharov (CKGZ) equations, the generalized Davey Sterwatson (DS) equations, the DS equations, the generalized Zakharov equations can be reduced to the same elliptic-like equation. Then, by using the generally projective Riccati equation method, many kinds of exact solutions of the above mentioned equations are obtained in a unified way. These solutions include new solitary wave, periodic and rational solutions. 1

3 1 INTRODUCTION In recent years, seeking exact solutions of nonlinear partial differential equations (NLPDEs) is of great significance as it appears that these NLPDEs are mathematical models of complex physics phenomena arising in physics, mechanics, biology, chemistry and engineers. In order to help engineers and physicists to better understand the mechanism that governs these physical models or to better provide knowledge to the physical problem and possible applications, a vast variety of the powerful and direct methods have been derived. Among these are the inverse scattering method [1], Hirota s method[2], Bäcklund transformation [3,4] tanh-function method [5-8], extended tanh-function method [9-12]. In the literature [13], Conte et al. presented a general ansatz to seek more new solitary wave solutions of some NLPDEs that can be expressed as a polynomial in two elementary functions which satisfy a projective Riccati equations [14]. More recently, Yan developed conte s method and presented the general projective Riccati equation method [15]. Several authors used the Yan s technique to solve many NLPDEs [ 16-19]. In this paper, we will apply the general projective Riccati equation method to solve some class of NLPDEs The key idea of this method is to introduce a new projective Riccati equation and use its solutions to replace the elementary functions in the projective Riccati equation method [13], which simply proceeds as follows [15-19] Step 1. For a given NLPDE in the unknown u(x,y,z,...,t) which are solutions of the ordinary differential equation (ODE) E(u, u, u,...) = 0 obtained by the travelling wave reduction u(x, y, z,..., t) u(ξ = λ 1 x + λ 2 y + λ 3 z λ n t). τ(ξ). Then we seek solu- Step 2. We introduce two new variables σ(ξ), tions of u(ξ) in the following form T ype I when R 0 n u(ξ) = σ i 1 (ξ)[a i σ(ξ) + B i τ(ξ)] + A 0, (1) i=1 where A i and B i are constants to be determined later. σ(ξ), τ(ξ) are solutions of the following new projective Riccati equation σ (ξ) = ɛσ(ξ)τ(ξ), τ (ξ) = R+ɛτ 2 (ξ) mσ(ξ), ɛ = ±1. (2.a) 2

4 where m, R are constants. It is easy to see that Eq.(2.a) admits the first integral with R 0 [ ] τ 2 (ξ) = ɛ R 2m σ(ξ) + m2 +r i R σ2 (ξ), r i = ±1 (2.b) or we seek solutions of u(ξ) in the following form Type II when R=m=0, where τ satisfies u(ξ) = n A i τ i (ξ) + A 0, i=1 τ (ξ) = τ 2 (ξ) (3.a) (3.b) The parameter n in (1) and (3.a) can be determined by balancing the highestorder derivative term in the E(u, u, u,...) = 0. Step 3. Substituting system (1) along with condition (2.a) and (2.b) (or (3.a) along with (3.b)) into the E(u, u, u,...) = 0 and setting the coefficient of σ i τ j (j=0,1, i=0,1,2,3,...) (τ l, l=0, 1,...) to zero yields a set of overdetermined algebraic equations, from which the constants A i, B i, R, m, and λ i (A i, and λ i ) can be found explicitly. Step 4. We know that (2.a) and (2.a) admit the following solutions Case 1 when ɛ = 1, r i = 1, R 0 σ 1 (ξ) = R sech( Rξ) m sech( Rξ)+1, τ 1(ξ) = Case 2 when ɛ = 1, r i = 1, R 0 R tanh( Rξ) m sech( Rξ)+1, (4) σ 2 (ξ) = R csch( Rξ) m csch( Rξ)+1, τ 2(ξ) = Case 3 when ɛ = 1, r i = 1, R 0 σ 3 (ξ) = R sec( Rξ) m sec( Rξ)+1, τ 3(ξ) = R coth( Rξ) m csch( Rξ)+1, (5) R tan( Rξ) m sec( Rξ)+1, (6) σ 4 (ξ) = R csc( Rξ) m csc( Rξ)+1, R cot( Rξ) τ 4(ξ) = m csc(. (7) Rξ)+1 3

5 Case 4 when R = m = 0, σ 5 (ξ) = C ξ = Cɛτ 5(ξ), τ 5 (ξ) = 1 ɛξ. (8) C is a constant. Substituting the constants A i, B i, R, m and λ i (A i and λ i ) obtained in step 2 into (1) along with (4)-(7) (into (3.a) along with (8)) to obtain soliton and periodic ( rational) solutions of the NLPDE in concern. We will continue by showing that we can use the above described method to present exact solutions to some class of NLPDEs which can be transformed to the same family of elliptic-like equation. 2 Exact solutions of some class of NLPDEs 2.1 A new Hamiltonian amplitude equation A new Hamiltonian amplitude equation iu x + u tt + 2η u 2 u βu xt = 0, (9) where η = ±1, β << 1 was recently introduced by Wadati et al [20]. This equation governs certain instabilities of modulated wave trains, and the addition of the term βu xt overcomes the ill-posedness of the unstable nonlinear Schrȯdinger equation. Let u(x, t) = φ(ξ)e i(kx Ωt), ξ = px ωt. (10) Substituting Eq.(10) into Eq.(9), we have (ω 2 +pβω)φ (ξ) +i(p+2ωω+kβω+pβω)φ (ξ) (K+Ω 2 +pkβω)φ(ξ)+2ηφ 3 (ξ) = 0. (11) If we take p(1 + βω) ω = 2Ω + Kβ. (12) Eq.(11) is transformed into the following one 4

6 where φ (ξ) + k 1 φ(ξ) + φ 3 (ξ) = 0, (13.a) k 1 = K+Ω2 +βkω ω 2 +pβω, = 2η ω 2 +pβω. (13.b) Then the solutions of (9) are given by u(x, t) = φ(ξ)e i(kx Ωt), where φ(ξ) is defined by (54)-(69) and the other parameters are determined by ξ = px ωt, ω = p(1+βω), k 2Ω+Kβ 1 = K+Ω2 +βkω, k ω 2 +pβω 3 = 2η. ω 2 +pβω 2.2 a nonlinear wave equation Consider the nonlinear wave equation in Rev. [21] u tt + αu xx + βu + γu 3 = 0, (14) where α, β and γ are constants. Eq.(14) contains some particular important equations such as Duffing, Klein-Gordon and Landau=Ginzburg-Higgs equation. We assume that Eq.(14) has exact solution in the form u(x, t) = φ(ξ), ξ = px ωt. (15) Substituting Eq.(15) into Eq.(14), we have (ω 2 + αp 2 )φ (ξ) + βφ(ξ) + γφ 3 (ξ) = 0. (16) Then Eq.(16) can be written as φ (ξ) + k 1 φ(ξ) + φ 3 (ξ) = 0, (17.a) where k 1 = β ω 2 +αp 2, = γ ω 2 +αp 2. (17.b) Then the solutions of (14) are given by u(x, t) = φ(ξ), where φ(ξ) is defined by (54)-(69), and the other parameters are expressed by ξ = px ωt, k 1 = β, k ω 2 +αp 2 3 = γ. ω 2 +αp 2 5

7 2.3 Coupled Klein-Gordon-Zakharov equations The coupled nonlinear Klein-Gordon-Zakarov equations [1] read u tt c u + f0 2 u + δuv = 0, v tt c v β 2 u 2 = 0. (18) We seek its following wave packet solution u(x, y, z, t) = φ(ξ)e i(kx+ly+nz Ωt), v(x, y, z, t) = v(ξ), xi = px + qy + rz ωt, (19) where both φ(ξ) and v(ξ) are real function. Substituting Eq.(19) into Eq.(18) yields (ω 2 c 2 0P 2 )φ (ξ) + 2i(ωΩ c 2 0KP)φ (ξ) (ω 2 c 2 0K 2 f 2 0 )φ(ξ) + δvφ(ξ) = 0 (ω 2 c 2 0P 2 )v (ξ) βp 2 (φ 2 (ξ)) = 0, (20) where K =(k, l, n), K 2 = k 2 + l 2 + n 2, P=(p, q, r), P 2 =p 2 + q 2 + r 2 ; K.P=kp + lq + nr. (21) If we take ωω = c 2 0K.P, (22) then (20) is reduced to (ω 2 P 2 c 2 0)φ (ξ) (ω 2 K 2 c 2 0 f 2 0 )φ(ξ) + δvφ(ξ) = 0, (23.a) (ω 2 P 2 c 2 0)v (ξ) βp 2 (φ 2 (ξ)) = 0. (23.b) Integrating (23.b) once with respect to ξ, we get 6

8 (ω 2 P 2 c 2 0)v (ξ) βp 2 φ 2 (ξ) = c. (24) where c is integration constant. Because we find the special form of exact solutions for simplicity purpose, we take c = 0 and integrating this formula once again, we have v(ξ) = C ω 2 c 2 0P 2 + βp 2 ω 2 c 2 0P 2 φ2 (ξ), (25) where C is an integration constant. Substituting (25) into (23.a) yields (ω 2 c 2 0P 2 ) 2 φ (ξ)+[(ω 2 c 2 0P 2 )( ω 2 +c 2 0K 2 c 2 0+f 2 0 )+δc]φ(ξ)+δβp 2 φ 3 (ξ) = 0, (26) Eq.(26) can be expressed as φ (ξ) + k 1 φ(ξ) + φ 3 (ξ) = 0, (27.a) where k 1 = [(ω2 c 2 0 P2 )( ω 2 +c 2 0 K2 +f 2 0 )+δc] (ω 2 c 2 0 P2 ) 2, = δβp2 (ω 2 c 2 0 P2 ) 2. (27.b) Then the solutions of CKGZ equations are given by u(x, y, z, t) = φ(ξ) e i(kx+ly+nz Ωt) C, v(x, y, z, t) = + βp2 φ 2 (ξ), where φ(ξ) intervening ω 2 c 2 0 P2 ω 2 c 2 0 P2 in these solutions is given by (54)-(69), the other parameters are defined by Ω = c2 0 K.P, ξ = px + qy + rz ωt, k ω 1 = [(ω2 c 2 0 P2 )( ω 2 +c 2 0 K2 +f0 2)+δC], k (ω 2 c 2 0 P2 ) 2 3 = δβp 2 (ω 2 c 2 0 P2 ) A class of NLPDEs with constant coefficients We consider a class of NLPDEs with constant coefficients [22] iu t + µ(u xx + D 1 u yy ) + E 1 u 2 u + C 1 un = 0, D 2 n tt + (n xx E 2 u yy ) + C 2 ( u 2 ) xx = 0, (28.a) (28.b) where µ, D i, E i, C i (i=1,2) are real constants and µ 0, D 1 0, C 1 0, C 2 0. Eqs.(28.a), (28.b) are a class of physically important equation. In fact, if one takes 7

9 µ = 1 2 κ2, D 1 = 2µ, E 1 = α, C 1 = 1, D 2 = 0, E 2 = D 1, C 2 = 2α, κ 2 = ±1 (29) Then Eqs.(28.a), (28.b) represent the Davey-Sterwatson (DS) equations [23] iu t κ2 (u xx + κ 2 u yy ) + α u 2 u un = 0, n xx κ 2 n yy 2α( u 2 ) xx = 0. (30.a) (30.b) If one takes n = n(x, t) i.e. n y = 0, µ = 1, D 1 = 0, E 1 = 2λ, E 2 = 1, C 2 = 1, C 1 = 2. (31) Then Eqs.(28a) and (28.b) become generalized Zakharov (GZ) equations [24] iu t + u xx 2λ u 2 u + 2un = 0, n tt n xx + ( u 2 ) xx = 0. Since u is a complex function, we assume that (32.a) (32.b) u(x, y, t) = φ(ξ)e i(kx+ly Ωt), v(x, y, t) = v(ξ), ξ = px+qy ωt (33) where both φ(ξ) and v(ξ) are real functions, k, l, p, q, Ω and ω are constants to be determined later. Substituting Eq.(33) into Eqs.(28.a) and (28.b), we have the following ODE for φ(ξ) and v(ξ) µ(p 2 + D 1 q 2 )φ (ξ) + [Ω µ(k 2 + D 1 l 2 )]φ(ξ) + E 1 φ 3 (ξ) + i[ ω + 2µ(kp + D 1 lq)]φ (ξ) + C 1 φ(ξ)v(ξ) = 0, (34.a) Setting (D 2 ω 2 + p 2 E 2 q 2 )v (ξ) + C 2 p 2 (φ 2 (ξ)) = 0. (34.b) Then (34.a) and (34.b) reduce to ω = 2µ(kp + D 1 lq). (35) µ(p 2 +D 1 q 2 )φ (ξ)+[ω µ(k 2 +D 1 l 2 )]φ(ξ)+e 1 φ 3 (ξ)+c 1 φ(ξ)v(ξ) = 0, (36.a) 8

10 (D 2 ω 2 + p 2 E 2 q 2 )v (ξ) + C 2 p 2 (φ 2 (ξ)) = 0. (36.b) Integrating (36.b) once, we get (D 2 ω 2 + p 2 E 2 q 2 )v (ξ) + C 2 p 2 (φ 2 (ξ)) = C, (37) where C is integration constant, then we take C = 0 and integrating the formula once again, we have v(ξ) = C D 2 ω 2 + p 2 E 2 q 2 C 2 p 2 D 2 ω 2 + p 2 E 2 q 2 φ2 (ξ). (38) Substituting (38) into (36.a) yields µ(p 2 +D 1 q 2 )(D 2 ω 2 +p 2 E 2 q 2 )φ (ξ)+[c 1 C (D 2 ω 2 +p 2 E 2 q 2 )(Ω µ(k 2 + D 1 l 2 ))]φ(ξ)+[e 1 (D 2 ω 2 +p 2 E 2 q 2 ) C 1 C 2 p 2 ]φ 3 (ξ) = 0, (39) Eq.(39) can be written as where φ (ξ) + k 1 φ(ξ) + φ 3 (ξ) = 0, (40.a) k 1 = C 1C (D 2 ω 2 +p 2 E 2 q 2 )(Ω µ(k 2 +D 1 l 2 )) µ(p 2 +D 1 q 2 )(D 2 ω 2 +p 2 E 2 q 2 ), = E 1(D 2 ω 2 +p 2 E 2 q 2 ) C 1 C 2 p 2 µ(p 2 +D 1 q 2 )(D 2 ω 2 +p 2 E 2 q 2 ). (40.b) Then the solutions of the GDS equations can be written as u(x, y, t) = φ(ξ)e i(kx+ly Ωt) C C, v(x, y, t) = D 2 ω 2 +p 2 E 2 2 p 2 q 2 D 2 ω 2 +p 2 E 2 φ 2 (ξ), φ(ξ) intervening in these solutions are given by (54)-(69), the other parameters are q 2 defined by ξ = px + qy ωt, ω = 2µ(kp + D 1 lq) and k 1 and satisfy (40.b) We may obtain from (30) that v(x, y, t) = u(x, y, t) = φ(ξ)e i(kx+ly Ωt), C p 2 κ 2 q + 2αp2 2 p 2 κ 2 q 2 φ2 (ξ), (41 ) ξ = px + qy ωt, ω = κ 2 (kp + κ 2 lq), φ (ξ) + k 1 φ(ξ) + φ 3 (ξ) = 0, (41.a) 9

11 where k 1 = 2C+(p2 κ 2 q 2 )( 2Ω+κ 2 (k 2 +κ 2 l 2 )) κ 2 (p 2 +κ 2 q 2 )(κ 2 q 2 p 2 ), = 2α κ 2 (κ 2 q 2 p 2 ). (41.b) Then from (32) we have that v(x, t) = u(x, t) = φ(ξ)e i(kx Ωt), C p 2 ω + p2 2 p 2 ω 2 φ2 (ξ), (42 ) ξ = px ωt, ω = 2kp, where φ (ξ) + k 1 φ(ξ) + φ 3 (ξ) = 0, (42.a) k 1 = 2C (p2 ω 2 )(Ω k 2 )) p 2 (p 2 ω 2 ), = 2(p2 λ(p 2 ω 2 )) p 2 (p 2 ω 2 ). (42.b) Where for the solutions of the DS equations and the GZ equations defined above the φ(ξ) appearing in them is determined by (54)-(69). 2.5 Exact solutions of the elliptic-like equation Now let us return back to Eqs.(13), (17), (27), (40), (41) and (42). According to step 2 in section 1, by balancing the higher-order derivative term φ with the nonlinear term φ 3, w get n=1 in (1). Therefore we suppose that Eqs.(13), (17), (27), (40), (41) and (42) have the following formal solutions with R 0 φ(ξ) = A 0 + A 1 σ(ξ) + B 1 τ(ξ), (43) where A 0, A 1 and B 1 are constants to be determined later. σ(ξ) and τ(ξ) satisfy (2.a) and (2.b). According to the step 3, we substitute (43) into the elliptic-like equations (13), (17), (27), (40), (41) and (42) along with (2.a) and (2.b). With the aid of Mathematica, collecting all terms with the same power in σ j (ξ)τ i (ξ), j = 0, 1, 2, 3, 4; i = 0, 1 and setting the coefficients of these terms σ j (ξ)τ i (ξ) to zero yields a set of over-determined algebraic 10

12 equations with respect to A 0, A 1, B 1, R, and m. Const : k 1 A 0 + [A 3 0 3ɛRA 0 B 2 1] = 0, (44) σ(ξ) : k 1 A 1 + [3A 2 0A 1 3ɛRA 1 B 2 1+6ɛA 0 B 2 1m] ɛa 1 R = 0, (45) τ(ξ) : k 1 B 1 + [3A 2 0B 1 ɛrb 3 1] = 0, (46) σ(ξ)τ(ξ) : [6A 0 A 1 B 1 + 2ɛmB 3 1] + ɛmb 1 = 0, (47) σ 2 (ξ) : [3A 2 0A 1 +6ɛmA 1 B 2 1 3ɛ R A 0B 2 1(m 2 +r i )]+3ɛmA 1 = 0, (48) σ 2 (ξ)τ(ξ) : [3A 2 1B 1 ɛ B3 1 R (m2 +r i )] 2ɛ m2 +r i R B 1 = 0, (49) σ 3 (ξ) : [A 3 1 3ɛ m2 +r i A R 1B1] 2 2ɛ m2 +r i A R 1 = 0. (50) From which using Mathematica and Wu elimination Method [25,26], we get the following results (I) : A 0 = A 1 = m = 0, R = k 1 ɛ, A 2 1 = 2r i k 1, ɛ = ±1, r i = ±1, (51) (II) : A 0 = A 1 = m = 0, R = k 1 2ɛ, B2 1 = 2, ɛ = ±1, r i = ±1, (52) (III) : A 0 = 0, R = 2k 1, A 2 ɛ 1 = m2 +r i 4k 1 B1 2 = 1 2, ɛ = ±1, r i = ±1, (53) Therefore, from (4)-(7) and (51)-(53), we obtain fifteen kinds of exact travelling wave solutions of Eqs.(13), (17), (27), (40), (41) and (42). Family 1. Dark soliton solutions φ 1 (ξ) = ± k 1 tanh( Family 2. Singular dark soliton solutions φ 2 (ξ) = ± k 1 coth( k1 2 k1 2 ξ), (54) ξ), (55) 11

13 Family 3. Bright soliton solutions φ 3 (ξ) = ± 2 sech( k 1 ξ), (56) Family 4. Singular soliton solutions 2 φ 4 (ξ) = ± csch( k 1 ξ), (57) Family 5. Periodic wave solutions k1 φ 5 (ξ) = ± tan( Family 6. Periodic wave solutions k1 φ 6 (ξ) = ± cot( Family 7. Periodic wave solutions k 1 2 k 1 2 ξ), (58) ξ), (59) φ 7 (ξ) = ± 2 sec( k 1 ξ), (60) Family 8. Periodic wave solutions φ 8 (ξ) = ± 2 csc( k 1 ξ), (61) Family 9. combined formal soliton-like solutions φ 9 (ξ) = ± k 1 [ 1 + m2 sech( 2k 1 ξ) m sech( 2k 1 ξ) + 1 Family 10. combined formal soliton-like solutions + ɛ tanh( ] 2k 1 ξ) m sech(, (62) 2k 1 ξ) + 1 φ 10 (ξ) = ± k 1 [ 1 + m2 csch( 2k 1 ξ) m csch( 2k 1 ξ) ɛ coth( ] 2k1 ξ) m csch(, (63) 2k 1 ξ)

14 Family 11. combined formal periodic wave-like solutions φ 11 (ξ) = ± k1 [ 1 m2 sec( 2k 1 ξ) m sec( 2k 1 ξ) ɛ tan( ] 2k1 ξ) m sec(, (64) 2k 1 ξ) + 1 Family 12. combined formal periodic wave-like solutions φ 12 (ξ) = ± k1 [ 1 m2 csc( ] 2k 1 ξ) m csc( 2k 1 ξ) ɛ cot( 2k1 ξ) m csc(, (65) 2k 1 ξ) + 1 When m = ±1 then we have the following solutions Family 13. new soliton solutions φ 13 (ξ) = ± Family 14. new periodic wave solutions k 1 tanh( 2k1 ξ) ɛ sech( 2k 1 ξ) + 1, (66) φ 14 (ξ) = ± k 1 tan( 2k1 ξ) ɛ sec( 2k 1 ξ) + 1, (67) Family 15. new periodic wave solutions φ 15 (ξ) = ± k 1 cot( 2k1 ξ) ɛ csc( 2k 1 ξ) + 1. (68) According to the above mentioned method in section 1 and (3.a) and (3.b), we assume that Eqs. (13), (17), (27), (40), (41), (42) have the solutions in the form φ(ξ) = A 0 + A 1 τ(ξ), when R=m=0, then we obtain the rational solutions. Family 16. Rational solutions: where k 1 = 0, A 0 = 0. φ 16 (ξ) = ± 2 1 ξ, (69) 13

15 3 Conclusion In this paper, by using a more general transformation (1) and the general projective Riccati equations method, we have been able to obtain in a unified way, by the help of symbolic computation system-mathematica, many kinds of exact solutions to a class of NLPDEs. This class of NLPDEs is characterized by the fact that it can be reduced through a simple transformation to the elliptic-like equation φ (ξ) + k 1 φ(ξ) + φ(ξ) 3 = 0. It is obvious that by using this simple transformation, the computation quantity evolved in solving nonlinear equations is greatly reduced. This method has proved its efficiency to the Hamiltonian amplitude equation, the nonlinear wave equation, the coupled Klein-Gordon-Zakharov (CKGZ) equations, the generalized Davey Sterwatson (GDS) equations, the DS equations, the generalized Zakharov (GZ) equations. To our knowledge, it s the first time that the general projective equations method is used for solving a system of coupled equations. Acknowledgements I will like to thank the Institute for Mathematics and its Applications of Minneapolis for the generous hospitality and support. I thank Professor George Sell for useful discussions. References [1] M. Ablowitz and P.A. Clarkson: Soliton, nonlinear evolution equations and inverse scattering. New York; Cambridge University Press [2] R. Hirota: Phys. Rev. Lett. 27 (1971) [3]M. Wadati: J. Phys. Soc. Jpn 38 (1975) 673. [4] M. Wadati: J. Phys. Soc. Jpn 38 (1975) 681. [5] W. Malfliet: Am. J. Phys. 60 (1992) 650. [6] E.J. Parkes and B.R. Duffy: Comput. Phys. 98 (1996) 288. [7] E.J. Parkes and B.R. Duffy: Phys. Lett. A 229 (1997) 217. [8] Y.T. Gao and B. Tian: Comput. Math. Appl. 33 (1997) 115. [9] E.G. Fan: Phys. Lett. A 277 (2000) 212. [10] E.G. Fan: Z. Naturforsch A 56 (2001) 312. [11] Z.Y. Yan: Phys. Lett. A 292 (2001) 100. [12] B. Li, Y. Chen and H.Q. Zhang: Chaos, Solitons and Fractals 15 (2003) 647. [13] R. Conte and M. Musette: J. Phys. A 25 (1992)

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