Envelope Periodic Solutions to Coupled Nonlinear Equations

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1 Commun. Theor. Phys. (Beijing, China) 39 (2003) pp c International Academic Publishers Vol. 39, No. 2, February 15, 2003 Envelope Periodic Solutions to Coupled Nonlinear Equations LIU Shi-Da, 1,2 FU Zun-Tao, 1,2 LIU Shi-Kuo, 1 and ZHAO Qiang 1 1 School of Physics, Peking University, Beijing , China 2 State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing , China (Received June 10, 2002) Abstract The envelope periodic solutions to some nonlinear coupled equations are obtained by means of the Jacobi elliptic function expansion method. And these envelope periodic solutions obtained by this method can degenerate to the envelope shock wave solutions and/or the envelope solitary wave solutions. PACS numbers: Jb Key words: Jacobi elliptic function, nonlinear coupled equations, envelope periodic solution 1 Introduction Numerous nonlinear models (nonlinear equations) are proposed to understand the physical mechanism in different physical problems. Generally speaking, the exact analytical solutions to these nonlinear equations are hardly obtained, and many numerical methods have to be applied to solve these nonlinear equations. Fortunately, a few exact analytical solutions for some nonlinear equations can be found under certain conditions, for example, the solitary wave solutions, shock wave solutions and periodic solutions, and so on. Applying these analytical solutions, we can check the reliability of numerical solutions in the same control parameters. Much effort has been spent on the construction of exact solutions of nonlinear equations. A number of methods have been proposed, such as the homogeneous balance method, [1 3] the hyperbolic tangent function expansion method, [4 6] the nonlinear transformation method, [7,8] the trial function method, [9,10] and sine-cosine method. [11] These methods, however, can only obtain the shock and solitary wave solutions or the periodic solutions with the elementary functions, [1 12] but cannot get the generalized periodic solutions of nonlinear equations. Although Porubov et al. [13 15] have obtained some periodic solutions to some nonlinear equations, they used the Weierstrass elliptic function and involved complicated deduction. The Jacobi elliptic function expansion method [16,17] has been proposed and applied to obtain the generalized periodic solutions to some nonlinear equations and their corresponding shock wave and solitary wave solutions. In this paper, this method is applied to get the envelope periodic solutions and corresponding envelope shock or solitary wave solutions to some coupled nonlinear equations. 2 Envelope Periodic Solutions for Coupled NLS Equations The coupled nonlinear Schrödinger (NLS) equations read i u t + α 2 u x 2 + [β 1 u 2 + β 2 v 2 ]u = 0, i v t + α 2 v x 2 + [β 1 v 2 + β 2 u 2 ]v = 0. (1) We seek the following wave packet solutions Substituting Eq. (2) into Eqs. (1) yields Taking 2αk = c g, ω αk 2 = ( > 0), we have u = φ(ξ) e i(kx ωt), v = ψ(ξ) e i(kx ωt), ξ = p(x c g t). (2) dξ 2 + ip(2αk c g) dφ dξ + (ω αk2 )φ + β 1 φ 3 + β 2 φψ 2 = 0, αp 2 d2 ψ dξ 2 + ip(2αk c g) dψ dξ + (ω αk2 )ψ + β 1 ψ 3 + β 2 φ 2 ψ = 0. (3) dξ 2 φ + β 1φ 3 + β 2 φψ 2 = 0, αp 2 d2 ψ dξ 2 ψ + β 1ψ 3 + β 2 φ 2 ψ = 0. (4) The project supported by National Natural Science Foundation of China (No ) and the State Key Project of National Natural Science Foundation of China (No )

2 168 LIU Shi-Da, FU Zun-Tao, LIU Shi-Kuo, and ZHAO Qiang Vol. 39 Next, we solve Eq. (4) using Jacobi elliptic function expansions. 2.1 Jacobi Elliptic Sine Function Expansion By the Jacobi elliptic function expansion method, φ(ξ) and ψ(ξ) can be expressed as a finite series of Jacobi elliptic function, sn ξ, i.e., φ(ξ) = a j sn j ξ, ψ(ξ) = b j sn j ξ, (5) we can select n 1 and n 2 to balance the derivative term of the highest order and nonlinear term in Eq. (4) and get the ansatz solution of Eq. (4) in terms of sn ξ Substituting Eq. (6) into Eqs. (4), one can get φ(ξ) = a 0 + a 1 sn ξ, ψ(ξ) = b 0 + b 1 sn ξ. (6) ( + β 1 a β 2 b 2 0)a 0 + {[ + 3β 1 a 2 0 (1 + m 2 )αp 2 ]a 1 + β 2 (2a 0 b 1 + a 1 b 0 )b 0 }sn ξ + [3β 1 a 0 a β 2 (a 0 b 1 + 2a 1 b 0 )b 1 ]sn 2 ξ + (β 1 a αp 2 + β 2 b 2 1)a 1 sn 3 ξ = 0, ( + β 1 b β 2 a 2 0)a 0 + {[ + 3β 1 b 2 0 (1 + m 2 )αp 2 ]b 1 + β 2 (2a 1 b 0 + a 0 b 1 )a 0 }sn ξ + [3β 1 b 0 b β 2 (a 1 b 0 + 2a 0 b 1 )a 1 ]sn 2 ξ + (β 1 b αp 2 + β 2 a 2 1)b 1 sn 3 ξ = 0, (7) and then set the coefficients of the different powers for sn ξ to be zeros to obtain the algebraic equations about a j and b j, ( + β 1 a β 2 b 2 0)a 0 = 0, [ + 3β 1 a 2 0 (1 + m 2 )αp 2 ]a 1 + β 2 (2a 0 b 1 + a 1 b 0 )b 0 = 0, 3β 1 a 0 a β 2 (a 0 b 1 + 2a 1 b 0 )b 1 = 0, (β 1 a αp 2 + β 2 b 2 1)a 1 = 0, ( + β 1 b β 2 a 2 0)a 0 = 0, [ + 3β 1 b 2 0 (1 + m 2 )αp 2 ]b 1 + β 2 (2a 1 b 0 + a 0 b 1 )a 0 = 0, 3β 1 b 0 b β 2 (a 1 b 0 + 2a 0 b 1 )a 1 = 0, (β 1 b αp 2 + β 2 a 2 1)b 1 = 0, (8) from which the coefficients a j and b j and corresponding constraints can be determined as a 0 = b 0 = 0, p 2 = (1 + m 2 )α, a 1 = b 1 = ± 2 αp 2. (9) β 1 + β 2 Then the envelope periodic solutions for u and v are 2 (1 + m 2 )(β 1 + β 2 ) sn (1 + m 2 )α (x c gt) e i(kx ωt) (10) and when m 1, corresponding envelope solitary wave solutions are tanh β 1 + β 2 2α (x c gt) e i(kx ωt). (11) 2.2 Jacobi Elliptic Cosine Function Expansion φ(ξ) and ψ(ξ) can be expressed as a finite series of Jacobi cosine elliptic function, cn ξ, i.e., φ(ξ) = a j cn j ξ, ψ(ξ) = b j cn j ξ, (12) and the ansatz solution of Eq. (4) in terms of cn ξ is φ(ξ) = a 0 + a 1 cn ξ, ψ(ξ) = b 0 + b 1 cn ξ. (13) Similarly, another kind of envelope periodic solutions can be obtained, for cn ξ, which are 2 (β 1 + β 2 )( 2 1) cn α( 2 1) (x c gt) e i(kx ωt), (14)

3 No. 2 Envelope Periodic Solutions to Coupled Nonlinear Equations 169 and the limited solutions are 2 sech β 1 + β 2 α (x c gt) e i(kx ωt). (15) 2.3 Jacobi Elliptic Function of the Third Kind Expansion φ(ξ) and ψ(ξ) can be expressed as a finite series of Jacobi elliptic function of the third kind, dn ξ, i.e., φ(ξ) = a j dn j ξ, ψ(ξ) = b j dn j ξ, (16) and the ansatz solution of Eq. (4) in terms of dn ξ is φ(ξ) = a 0 + a 1 dn ξ, ψ(ξ) = b 0 + b 1 dn ξ. (17) Similarly, another kind of envelope periodic solutions can be obtained, for dn ξ, which are 2 (β 1 + β 2 )(2 m 2 ) dn α(2 m 2 ) (x c gt) e i(kx ωt), (18) and the limited solutions are the same one as Eq. (15). 2.4 Jacobi Elliptic Function cs ξ Expansion φ(ξ) and ψ(ξ) can be expressed as a finite series of Jacobi elliptic function, cs ξ, i.e., φ(ξ) = a j cs j ξ, ψ(ξ) = b j cs j ξ, and the ansatz solution of Eq. (4) in terms of cs ξ is cs ξ = cn ξ sn ξ (19) φ(ξ) = a 0 + a 1 cs ξ, ψ(ξ) = b 0 + b 1 cs ξ. (20) Similarly, another kind of envelope periodic solutions can be obtained, for cs ξ, which are 2 (β 1 + β 2 )(2 m 2 ) cs α(2 m 2 ) (x c gt) e i(kx ωt), (21) and the limited solutions are 2 β 1 + β 2 csch α (x c gt) e i(kx ωt). (22) The above analysis shows that the expansion in terms of different Jacobi elliptic functions can lead to different envelope periodic solutions and envelope solitary wave solutions, and more new results can be got in these expansions. 3 Envelope Periodic Solutions for Coupled Nonlinear Schrödinger KdV Equations The coupled nonlinear Schrödinger KdV equations read We seek its following wave packet solutions where both φ(ξ) and u(ξ) are real functions. Substituting Eq. (24) into Eqs. (23) yields u t + u u x + β 3 u x 3 v 2 x = 0, i v t + α 2 v δuv = 0. (23) x2 u = u(ξ), v = φ(ξ) e i(kx ωt), ξ = p(x c g t), (24) c g du dξ + u du dξ + βp2 d3 u dξ 3 dφ2 dξ = 0, dξ 2 + ip(2αk c g) dφ dξ + (ω αk2 )φ δuφ = 0. (25a) (25b)

4 170 LIU Shi-Da, FU Zun-Tao, LIU Shi-Kuo, and ZHAO Qiang Vol. 39 Integrating Eq. (25a) once and taking integration constant as zero, setting c g = 2αk and ω αk 2 = in Eq. (25b), then equations (25a) and (25b) take the following forms c g u u2 + βp 2 d2 u dξ 2 φ2 = 0, φ δuφ = 0. (26) dξ2 By the Jacobi elliptic function expansion method, φ(ξ) and u(ξ) can be expressed as a finite series of Jacobi elliptic function, sn ξ, i.e., u(ξ) = a j sn j ξ, φ(ξ) = b j sn j ξ. (27) We can select n 1 and n 2 to balance the derivative term of the highest order and nonlinear terms in Eq. (26) and get the ansatz solution of Eq. (26) in terms of sn ξ, It is obvious that there are the following formula u(ξ) = a 0 + a 1 sn ξ + a 2 sn 2 ξ, φ(ξ) = b 0 + b 1 sn ξ + b 2 sn 2 ξ. (28) d 2 u dξ 2 = 2a 2 (1 + m 2 )a 1 sn ξ 4(1 + m 2 )a 2 sn 2 ξ + 2 a 1 sn 3 ξ + 6m 2 a 2 sn 4 ξ, d 2 φ dξ 2 = 2b 2 (1 + m 2 )b 1 sn ξ 4(1 + m 2 )b 2 sn 2 ξ + 2 b 1 sn 3 ξ + 6m 2 b 2 sn 4 ξ, u 2 = a a 0 a 1 sn ξ + (2a 0 a 2 + a 2 1)sn 2 ξ + 2a 1 a 2 sn 3 ξ + a 2 2sn 4 ξ, φ 2 = b b 0 b 1 sn ξ + (2b 0 b 2 + b 2 1)sn 2 ξ + 2b 1 b 2 sn 3 ξ + b 2 2sn 4 ξ, uφ = a 0 b 0 + (a 0 b 1 + a 1 b 0 )sn ξ + (a 0 b 2 + a 1 b 1 + a 2 b 0 )sn 2 ξ + (a 1 b 2 + a 2 b 1 )sn 3 ξ + a 2 b 2 sn 4 ξ. (29) Then substituting Eqs. (28) and (29) into Eq. (26) leads to [ c g a 0 + a 2 0/2 + 2βp 2 a 2 b 2 0] + [ c g a 1 + a 0 a 1 βp 2 (1 + m 2 )a 1 2b 0 b 1 ]sn ξ + [ c g a 2 + (2a 0 a 2 + a 2 1)/2 4βp 2 (1 + m 2 )a 2 (2b 0 b 2 + b 2 1)]sn 2 ξ + [a 1 a βp 2 a 1 2b 1 b 2 ]sn 3 ξ + [a 2 2/2 + 6m 2 βp 2 a 2 b 2 2]sn 4 ξ = 0, [2αp 2 b 2 b 0 δa 0 b 0 ] + [ αp 2 (1 + m 2 )b 1 b 1 δ(a 0 b 1 + a 1 b 0 )]sn ξ + [ 4αp 2 (1 + m 2 )b 2 b 2 δ(a 0 b 2 + a 1 b 1 + a 2 b 0 )]sn 2 ξ + [2αp 2 m 2 b 1 δ(a 1 b 2 + a 2 b 1 )]sn 3 ξ + [6αp 2 m 2 b 2 δa 2 b 2 ]sn 4 ξ = 0. (30) Setting the coefficients of each power of sn ξ to be zero, one can get the algebraic equations about a i and b i (i = 0, 1, 2), i.e., c g a 0 + a 2 0/2 + 2βp 2 a 2 b 2 0 = 0, c g a 1 + a 0 a 1 βp 2 (1 + m 2 )a 1 2b 0 b 1 = 0, c g a 2 + (2a 0 a 2 + a 2 1)/2 4βp 2 (1 + m 2 )a 2 (2b 0 b 2 + b 2 1) = 0, a 1 a βp 2 a 1 2b 1 b 2 = 0, a 2 2/2 + 6m 2 βp 2 a 2 b 2 2 = 0, 2αp 2 b 2 b 0 δa 0 b 0 = 0, αp 2 (1 + m 2 )b 1 b 1 δ(a 0 b 1 + a 1 b 0 ) = 0, 4αp 2 (1 + m 2 )b 2 b 2 δ(a 0 b 2 + a 1 b 1 + a 2 b 0 ) = 0, 2αp 2 m 2 b 1 δ(a 1 b 2 + a 2 b 1 ) = 0, 6αp 2 m 2 b 2 δa 2 b 2 = 0, (31) from which one can determine the coefficients a 2 = 6m2 αp 2 δ α a 0 = 2(α + βδ) b 0 = ± 1 2δ, b 2 = ±6m 2 p 2 α2 + 2αβδ 2δ 2, a 1 = b 1 = 0, { c g + 4βp 2 (1 + m 2 (α + 2βδ) [ ) 4p 2 (1 + m 2 ) + δ α 2δ 2 α 2 + 2αβδ [a 0 c g 4βp 2 (1 + m 2 )]. (32) ]},

5 No. 2 Envelope Periodic Solutions to Coupled Nonlinear Equations 171 So the periodic solutions for the coupled nonlinear Schrödinger KdV equations are u = a 0 + 6m2 αp 2 sn 2 [ p(x c g t)], δ (33a) v = b 0 ± 6m 2 p 2 α2 + 2αβδ 2δ 2 sn 2 [ p(x c g t)] e i(kx ωt), (33b) where a 0 and b 0 are given in Eqs. (32), and equation (33b) is envelope periodic solutions. When m 1, equations (33a) and (33b) degenerate to soliton and envelope soliton solution, respectively. Moreover, there are relations sn 2 ξ + cn 2 ξ = 1, m 2 sn 2 ξ + dn 2 ξ = 1, (34) so one can get the same results by using sn ξ, cn ξ or dn ξ expansion in solving the coupled nonlinear Schrödinger-KdV equations. 4 Envelope Periodic Solutions for Coupled Nonlinear Klein Gordon Schrödinger Equations The coupled nonlinear Klein Gordon Schrödinger equations read 2 u t 2 2 u c2 0 x 2 + f 0 2 u v 2 = 0, We seek its following wave packet solutions where both φ(ξ) and u(ξ) are real functions. Substituting Eq. (36) into Eqs. (35) yields i v t + α 2 v + βuv = 0. (35) x2 u = u(ξ), v = φ(ξ) e i(kx ωt), ξ = p(x c g t), (36) p 2 (c 2 g c 2 0) d2 u dξ 2 + f 2 0 u φ 2 = 0, dξ 2 + ip(2αk c g) dφ dξ + (ω αk2 )φ + βuφ = 0. Setting c g = 2αk and ω αk 2 = δ in Eq. (37b), then equations (37a) and (37b) take the following forms p 2 (c 2 g c 2 0) d2 u dξ 2 + f 2 0 u φ 2 = 0, (37a) (37b) δφ + βuφ = 0. (38) dξ2 By the Jacobi elliptic function expansion method, φ(ξ) and u(ξ) can be expressed as a finite series of Jacobi elliptic function sn ξ, i.e., u(ξ) = a j sn j ξ, φ(ξ) = b j sn j ξ. (39) We can select n 1 and n 2 to balance the derivative term of the highest order and nonlinear term in Eq. (38) and get the ansatz solution of Eq. (38) in terms of sn ξ, u(ξ) = a 0 + a 1 sn ξ + a 2 sn 2 ξ, φ(ξ) = b 0 + b 1 sn ξ + b 2 sn 2 ξ. (40) Similarly, substituting Eq. (40) into Eq. (38) will lead to the following results a 2 = 6m2 αp 2, b 2 = ±6m 2 p 2 α(c2 g c 2 0 ), a 1 = b 1 = 0, β β a 0 = α { f0 2β 1 2 p 4 (c 2 g c 2 0) [f 0 2 4(1 + m 2 )p 2 (c 2 g c 2 } 0)] 4(c 2 g c 2 0 ), b 0 = ± [f 2 0 4(1 + m 2 )p 2 (c 2 g c 2 0)] 2(c 2 g c 2 0 ) α(c2 g c 2 0 ) β So the periodic solutions for the coupled nonlinear Klein Gordon Schrödinger equations are u = α { f0 2β 1 2 p 4 (c 2 g c 2 0) [f 0 2 4(1 + m 2 )p 2 (c 2 g c 2 } 0)] 4(c 2 g c 2 0 ) 6m2 αp 2 sn 2 [ p(x c g t)], β. (41) (42a)

6 172 LIU Shi-Da, FU Zun-Tao, LIU Shi-Kuo, and ZHAO Qiang Vol. 39 v = ± [f 0 2 4(1 + m 2 )p 2 (c 2 g c 2 0)] 2(c 2 g c 2 0 ) α(c2 g c 2 0 ) ± 6m 2 p 2 α(c2 g c 2 0 ) sn 2 [ p(x c g t)] e i(kx ωt), β β where equation (42b) is envelope periodic solutions. When m 1, equations (42a) and (42b) degenerate to soliton and envelope soliton solution, respectively. 5 Conclusion In this paper, the exact envelope periodic solutions to some coupled nonlinear equations are obtained by use of Jacobi elliptic function expansion method. The envelope periodic solutions got by this method can degenerate as the envelope shock wave and envelope solitary wave solutions. Similarly, this solving process can be applied to other nonlinear equations, such as Landau Lifshitz equations, Kadomtsev Petviashvili equation and some others. (42b) References [1] M.L. WANG, Phys. Lett. A199 (1995) 169. [2] M.L. WANG, Y.B. ZHOU, and Z.B. LI, Phys. Lett. A216 (1996) 67. [3] L. YANG, Z. ZHU, and Y. WANG, Phys. Lett. A260 (1999) 55. [4] L. YANG, J. LIU, and K. YANG, Phys. Lett. A278 (2001) 267. [5] E.J. Parkes and B.R. Duffy, Phys. Lett. A229 (1997) 217. [6] E. FAN, Phys. Lett. A277 (2000) 212. [7] R. Hirota, J. Math. Phys. 14 (1973) 810. [8] N.A. Kudryashov, Phys. Lett. A147 (1990) 287. [9] M. Otwinowski, R. Paul, and W.G. Laidlaw, Phys. Lett. A128 (1988) 483. [10] S.K. LIU, Z.T. FU, S.D. LIU, and Q. ZHAO, Appl. Math. Mech. 22 (2001) 326. [11] C. YAN, Phys. Lett. A224 (1996) 77. [12] J.F. ZHANG, Chin. Phys. 8 (1999) 326. [13] A.V. Porubov, Phys. Lett. A221 (1996) 391. [14] A.V. Porubov and M.G. Velarde, J. Math. Phys. 40 (1999) 884. [15] A.V. Porubov and D.F. Parker, Wave Motion 29 (1999) 97. [16] S.K. LIU, Z.T. FU, S.D. LIU, and Q. ZHAO, Phys. Lett. A289 (2001) 69. [17] Z.T. FU, S.K. LIU, S.D. LIU, and Q. ZHAO, Phys. Lett. A290 (2001) 72.

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