Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to Nonlinear Schrödinger Equations with Variable Coefficient

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1 Commun Theor Phys (Beijing, China) 46 (006) pp c International Academic Publishers Vol 46, No 4, October 15, 006 Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to Nonlinear Schrödinger Equations with Variable Coefficient GE Jian-Ya, 1, WANG Rui-Min, 1, DAI Chao-Qing, and ZHANG Jie-Fang, 1 Normal Department of Humanities, College of Jinhua Professional Technology, Jinhua 31000, China Institute of Nonlinear Physics, Zhejiang Normal University, Jinhua 31004, China (Received December 6, 005; Revised April 5, 006) Abstract In this paper, by means of the variable-coefficient mapping method based on elliptical equation, we obtain explicit solutions of nonlinear Schrödinger equation with variable-coefficient These solutions include Jacobian elliptic function solutions, solitary wave solutions, soliton-like solutions, and trigonometric function solutions, among which some are found for the first time Six figures are given to illustrate some features of these solutions The method can be applied to other nonlinear evolution equations in mathematical physics PACS numbers: 0545Yv, 030Jr, 030Ik Key words: variable-coefficient mapping method based on elliptical equation, nonlinear Schrödinger equation, Jacobian elliptic function solutions, solitonic solutions, trigonometric function solutions 1 Introduction Optical solitons have a promising potential to become principal carriers in telecommunication because of their capability of propagating long distances without deformation 1 The optical soliton in a dielectric fiber was firstly proposed by Hasegawa and Tappert and verified experimentally by Mollenaur et al 3 The propagation of picosecond optical solitons in monomode fibers is governed by the celebrated nonlinear Schrödinger equation (NLSE) 3 A more general version of the NLSE is the variablecoefficient NLSE (VcNLSE), iψ t + a(t)ψ xx + b(t) ψ ψ = 0, (1) ψ = ψ(x, t) is a complex envelope of the electrical field in a comoving frame, x is the retarded time, t represents the distance along the direction of propagation (the longitudinal coordinate), a(t) is the group velocity dispersion (GVD) parameter, and b(t) is the Kerr nonlinearity parameter induced self-phase modulation (SPM) Equation (1) has been already studied by Grimshaw et al in Ref 4 When a = ±1/, b = 1, equation (1) reduces to the standard nonlinear Schrödinger equation 3 In the past few decades, there has been noticeable progress in the construction of the exact solutions of nonlinear partial differential equations (NLPDEs) With the help of exact solutions, when they exist, the phenomena modelled by these NLPDEs can be better understood They can also help to analyze the stability of these solutions and to check numerical analysis for these NLPDEs A vast variety of methods has been established to obtain exact solutions to a given nonlinear partial differential equation (NPDE), such as the inverse scattering method, 5 the Bäcklund transformation, 6 the tanh method, 7 the Jacobian elliptic function method, 8 the multilinear variable separation approach, 9 and the generalized Ricatti equation expansion method, 10 etc More recently, Wang et al 11 presented a new method, ie the mapping method based on elliptical equation, which is more simple and effective compared with the Jacobian elliptic function method 8 However, to our knowledge, most of the aforementioned methods are related to the constant-coefficient models though the study of variable-coefficient nonlinear equations has attracted much attention 1 because most of real nonlinear physical equations possess variable coefficients In this paper, we improved the mapping method based on elliptical equation to solve variable-coefficient nonlinear Schrödinger equation (1) and obtain several new exact soliton-like solutions for Eq (1) The paper is organized as follows In Sec, we briefly describe the variable-coefficient mapping method based on elliptical equation In Sec 3, several new exact soliton-like solutions for Eq (1) are obtained In particular, some new exact solitary wave solutions are given Finally, a short summary is presented Variable-Coefficient Mapping Method Based on Elliptical Equation Consider a nonlinear evolution equation, say, with two variables x and t, P (u, u x, u t, u xt, u xx, u tt, ) = 0 () Step 1 We assume that equation () has the following formal solutions: l u(x, t) = a 0 + a i F (ξ) + b i F (ξ) i, (3) i=1 a 0 = a 0 (x, t), a i = a i (x, t), b i = b i (x, t), (i = 1,, l), and ξ = ξ(x, t) are all arbitrary functions of The project supported by the Natural Science Foundation of Zhejiang Province of China under Grant No Y60531 Corresponding author, jf zhang@zjnucn

2 No 4 Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to 657 the indicated variables, and F (ξ) satisfies the following equation: 13 F (ξ) = c 0 + c F (ξ) + c 4 F 4 (ξ), (4) c 0, c, and c 4 are constants to be determined The solutions of Eq (4) with certain c 0, c, and c 4 are shown in Table 1, which are a series of Jacobian elliptic function solutions When modulus number m 0 or 1 (0 < m < 1), we can get trigonometric function solutions and hyperbolic function solutions Table 1 ODE and Jacobian elliptic function, relations between values of (c 0, c, c 4) and corresponding F (ξ) in F = c 0 + c F + c 4F 4 c 0 c c 4 F = c 0 + c F + c 4 F 4 F 1 (1 + m ) m F = (1 F )(1 m F ) sn ξ, cd ξ = cn ξ/dn ξ 1 m m 1 m F = (1 F )(m F + 1 m ) cn ξ m 1 m 1 F = (1 F )(F + m 1) dn ξ m (1 + m ) 1 F = (1 F )(m F ) ns ξ = (sn ξ) 1, dc ξ = dn ξ/cn ξ m m 1 1 m F = (1 F )(m 1)F m nc ξ = (cn ξ) 1 1 m m 1 F = (1 F )(1 m )F 1 nd ξ = (dn ξ) 1 1 m 1 m F = (1 + F )(1 m )F + 1 sc ξ = sn ξ/cn ξ 1 m 1 m (1 m ) F = (1 + m F )1 + (m 1)F sd ξ = sn ξ/dn ξ 1 m m 1 F = (1 + F )(F + 1 m ) cs ξ = cn ξ/sn ξ m (1 m ) m 1 1 F = (F + m )(F + m 1) ds ξ = dn ξ/sn ξ Step Determine the parameter l by balancing the highest order derivative terms with the nonlinear terms in Eq () It represents a subtle balance of the dissipation effect and the dispersion effect in physics soliton origins from Step 3 Substituting Eqs (3) and (4) into Eq () yields a set of algebraic polynomials for F (ξ) Eliminating all the coefficients of the powers of F (ξ) and c0 + c F + c 4 F 4 yields a series of differential equations, from which the functions a 0, a i, b i (i = 1,, l) and ξ are explicitly determined Step 4 Substituting a 0, a i, b i, ξ obtained in Step 3 into Eq (3), selecting the Jacobian elliptic functions with different parameters in Table 1, we can then derive all kinds of Jacobian elliptic function solutions of Eq () Remark 1 The significant difference between the method in Ref 11 and the above-mentioned method is that we replace the constants a 0, a i of transformation (11) in Ref 11 by differentiable functions a 0 (x, t), a i (x, t) Therefore, the algebraic equations are replaced by differential equations More importantly, we add the negative exponential power term b i F (ξ) i in new ansatz (3), so more types of solutions would be expected for some equations Moreover, the ξ(x, t) is an suitable function and is not always constrained to the travelling wave transformation 3 Exact Soliton-Like Solution of Eq (1) Now we turn to Eq (1) In order to obtain exact solution of Eq (1), firstly we make the transformation, ψ(x, t) = v(x, t) expiu(x, t), (5) v(x, t) and u(x, t) are amplitude and phase functions respectively Substituting Eq (5) into Eq (1) and separating the real and imaginary parts, thus vu t + a(t)(v xx vu x) + a(t)v 3 = 0, (6) v t + a(t)(v x u x + vu xx ) = 0 (7) Balancing the highest order derivative terms v xx with the nonlinear terms v 3 in Eq (6), we obtained l = 1 in Eq (3) Because it is difficult to search for the general solution like Eq (3) to nonlinear Schrödinger equation (1), we consider the following familiar ansatz: v(x, t) = f(t) + h(t)f (ξ) + g(t)f (ξ) 1, (8) ξ = p(t)x + q(t), u(x, t) = κ(t)x + Γ(t)x + Ω(t), (9) f(t), h(t), g(t), κ(t), Γ(t), and Ω(t) are functions of t to be determined, p(t) and q(t) are related to pulse width and group velocity respectively Furthermore, we assume that the phase u(x, t) has a quadratic form (9), ie there exists the chirped κ(t) term F (ξ) satisfies the value in Table 1 Substituting Eqs (8) and (9) into Eqs (6) and (7), eliminating x k F l (k = 0, 1,, l = 0, 1,, 3) and c0 + c F + c 4 F 4 yields g(4aκ + κ t ) = 0, (10) f(4aκ + κ t ) = 0, (11) h(4aκ + κ t ) = 0, (1) f(γ t + 4aΓκ) = 0, (13) h(γ t + 4aΓκ) = 0, (14) g(γ t + 4aΓκ) = 0, (15) ahp c 4 + bh 3 = 0, (16)

3 658 GE Jian-Ya, WANG Rui-Min, DAI Chao-Qing, and ZHANG Jie-Fang Vol 46 agc 0 p + bg 3 = 0, (17) 3bfh = 0, (18) 3bg f = 0, (19) haγ hap c 3hbf 3h bg + hω t = 0, (0) 3g bh gaγ + 3gbf + gap c gω t = 0, (1) fγ a + f 3 b fω t + 6fbgh = 0, () h(p t + 4apκ) = 0, (3) g(p t + 4apκ) = 0, (4) h t + ahκ = 0, (5) g(q t + apγ) = 0, (6) f t + afκ = 0, (7) h(q t + apγ) = 0, (8) (agκ + g t ) = 0 (9) Solving Eqs (10) (9) by means of Maple, we find f = 0, Γ = A 1 κ, p = A κ, h = A 3 κ (1/), c0 q = 1 A 1A κ + A 4, g = ± A 3 κ (1/), c 4 Ω = 4 (A 1 A c ) ± 3 c0 c 4 A κ + A 5, a = κ t b = c 4A κ t κ, (30) A 1 A 5 are arbitrary constants, κ(t) is an arbitrary function of t So the following formal solution can be obtained, ψ = A 3 κ 1/ c0 F (ξ) ± F (ξ) 1 c 4 expiκx + Γx + Ω, (31) ξ = A κx + A 1 A κ + A 4, Γ(t) = A 1 κ(t), 4 (A 1 A c ) ± 3 c0 c 4 A κ + A 5, b(t) = c 4A κ t Thus, several kinds of exact soliton-like solution of Eq (1) can be constructed as follows Family 1 F is sn-function, c 0 = 1, c = (1 + m ), c 4 = m, ψ 11 = A 3 sn(ξ) exp i A 1 x ((1 + m )A + A 1) (3) b(t) = a(t)a m (33) ψ 11 =A 3 tanh(ξ) exp i A 1 x (A +A 1) a(τ)dτ+a 5, b(t) = a(t)a From Eq (33), we see that the velocity of dark soliton is determined by A 1 a(t), phase shift is related to (A + A 1) a(τ)dτ, time shift is described by A1 a(τ)dτ, and the amplitude is determined by a(t)a /b(t) Therefore, we can select the parameters a(t) and b(t) to control the formation of dark soliton ψ 1 = A 3 sn(ξ) ± 1 m ns(ξ) expia 1 x + Ω, (34) Ω = (A c A 1) 6mA b(t) = a(t)a m ψ 1 = A 3 tanh(ξ) ± coth(ξ) expia 1 x + Ω, (35) Ω = (A c A 1) 6A, ψ 13 = A 3 κ 1/ sn(ξ) exp b(t) = a(t)a i κx + A 1 κx (A 1 + (1 + m )A )κ + A 5, (36) b(t) = m A κ t When m 1, the solitary wave solution is ψ 13 = A 3 κ 1/ tanh(ξ) exp i κx + A 1 κx (A 1 + A )κ + A 5, (37) b(t) = A κ t κ, ψ 14 = A 3 κ 1/ sn(ξ) ± 1 m ns(ξ) expiκx + Γx + Ω, (38) Γ(t) = A 1 κ(t), 4 (A 1 + A (1 + m )) ± 3 ma κ + A 5, b(t) = m A κ t

4 No 4 Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to 659 ψ 14 = A 3 κ 1/ tanh(ξ) ± coth(ξ) expiκx + Γx + Ω, (39) ξ = A κx + A 1 A κ + A 4, Γ(t) = A 1 κ(t), 4 (A 1 + A ) ± 3 A κ + A 5, b(t) = A κ t Family F is cn-function, c 0 = 1 m, c = m 1, c 4 = m, ψ 1 = A 3 cn(ξ) exp i A 1 x + (A (m 1) A 1) (40) a(τ)dτ +A 4, b(t) = a(t)a m ψ 1 = A 3 sech(ξ) exp i A 1 x + (A A 1) (41) b(t) = a(t)a From Eq (41), we see the velocity of bright soliton is determined by A 1 a(t), phase shift is related to (A A 1) a(τ)dτ, time shift is described by A1 a(τ)dτ, and the amplitude is determined by a(t)a /b(t) Therefore, we can select the parameters a(t) and b(t) to control the formation of bright soliton, 1 m ψ = A 3 cn(ξ)± m nc(ξ) expia 1 x+ω, (4) Ω = (A (m 1) A 1) 6 m (1 m )A b(t) = a(t)a m When m 1, we have the solution (41), ψ 3 = A 3 κ 1/ cn(ξ) exp i κx + A 1 κx (A 1 (m 1)A )κ + A 5, (43) b(t) = m A κ t ψ 3 = A 3 κ 1/ sech(ξ) exp i κx + A 1 κx (A 1 A )κ + A 5, (44) b(t) = A κ t ψ 4 = A 3 κ 1/ m cn(ξ) ± m nc(ξ) expiκx + Γx + Ω, (45) Γ(t) = A 1 κ(t), 4 (A 1 A (m 1)) ± 3 m (1 m )A κ + A 5, b(t) = m A κ t When m 1, we have the solution (44) Family 3 F is sc-function, c 0 = 1, c = m, c 4 = 1 m, ψ 31 = A 3 sc(ξ) exp i A 1 x + (A ( m ) A 1) (46) b(t) = a(t)a (1 m ) A 3 When m 0, the trigonometric function solution is ψ 31 = A 3 tan(ξ) exp i A 1 x + (A A 1) (47) b(t) = a(t)a A 3 1 ψ 3 = A 3 sc(ξ) ± 1 m cs(ξ) expia 1 x + Ω, (48) Ω = (A ( m ) A 1) 6 1 m A b(t) = a(t)a (1 m ) A 3 When m 0, the trigonometric function solution is ψ 3 = A 3 tan(ξ) ± cot(ξ) expia 1 x + Ω, (49)

5 660 GE Jian-Ya, WANG Rui-Min, DAI Chao-Qing, and ZHANG Jie-Fang Vol 46 Ω = (A A 1) 6A b(t) = a(t)a ψ 33 = A 3 κ 1/ sc(ξ) exp i κx + A 1 κx (A 1 ( m )A )κ + A 5, (50) b(t) = (1 m )A κ t When m 0, trigonometric function solution is ψ 33 = A 3 κ 1/ tan(ξ) exp i κx + A 1 κx (A 1 A )κ + A 5, (51) ξ = A κx + A 1 A κ + A 4, b(t) = A κ t ψ 34 = A 3 κ 1/ sc(ξ) ± 1 m cs(ξ) expiκx + Γx + Ω, (5) Γ(t) = A 1 κ(t), 4 (A 1 A ( m )) ± 3 1 m A κ + A 5, b(t) = (1 m )A κ t When m 0, trigonometric function solution is ψ 34 = A 3 κ 1/ tan(ξ)±cot(ξ) expiκx +Γx+Ω, (53) Γ(t) = A 1 κ(t), 4 (A 1 A ) ± 3 A κ + A 5, b(t) = A κ t Similarly, choosing c 0, c, c 4 from Table 1, we can get other 10 families of Jacobian elliptic function solutions and can obtain corresponding solitonic solutions or trigonometric function solutions For the limit of length, we do not list them here Remark Due to the arbitrariness of A 1 A 5, it is not difficult to verify that from the solutions (33) and (41) obtained by us, the solutions (39) and (4) in Ref 14 can be recovered But, to our knowledge, the other solutions, especially the Jacobian elliptic function solutions, were not reported before Now we discuss the standard nonlinear Schrödinger equation (NLSE), 4 iψ t ± 1 ψ xx + ψ ψ = 0, (54) which plays an important role in many physics context, eg nonlinear optics, plasmas physics, etc From Eqs (33) and (41), we have Dark soliton (normal dispersion region) ψ(x, t) = A tanha (x + A 1 t) + A 4 exp i A 1 x + 1 (A + A 1)t + A 5 (55) Bright soliton (anomalous dispersion region) ψ(x, t) = A secha (x + A 1 t) + A 4 exp i A 1 x + 1 (A A 1)t + A 5, (56) which are well known Fig 1 Plot of (a) U 1 = ψ 11, (b) U = ψ 11 with a(t) = sin(t), A 1 = A 3 = 1, A = 1/, A 4 = 0, m = 05 The more general soliton-like solutions obtained by variable-coefficient mapping method based on elliptical equation contain some arbitrary differential functions and some arbitrary constants, which can make one to discuss the behavior

6 No 4 Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to 661 of solutions as a function of these arbitrary differential functions and some arbitrary constants and also provide the enough freedom to construct solutions that may be related to the real physical problem As some illustrative samples, plots of ψ 11, ψ 11, ψ 13, ψ 1, ψ 1, ψ 3 with various parameters are shown in Figs 1 3 From Figs 1(b), (b), 3(a), and 3(b), we can see that the solutions obtained possess solitonic features Fig Plot of (a) U 3 = ψ 1, (b) U 4 = ψ 1 with a(t) = tanh (t), A 1 = A 3 = 1, A = 1/, A 4 = 0, m = 05 Fig 3 (a) Plot of U 5 = ψ 13, (b) U 6 = ψ 3 with κ(t) = sin(t), A 1 =, A = A 3 = 1, A 4 = 0 4 Summary and Discussion In this paper, by means of the variable-coefficient mapping method based on elliptical equation, we present explicit solutions of nonlinear Schrödinger equations with variable-coefficient These solutions include Jacobian elliptic function solutions, solitary wave solutions, soliton-like solutions, and trigonometric function solutions, among which some were not reported before Six figures are given to illustrate some features of these solutions In fact, we naturally present a more general ansats, which reads l u(x, t) = a 0 + a i F + b i F i + f i F i 1 c j F j + k i F i c j F j, (57) i=1 a 0 = a 0 (x, t), a i = a i (x, t), b i = b i (x, t), f i = f i (x, t), k i = k i (x, t), (i = 1,, l) and ξ = ξ(x, t) are differentiable functions which need to be determined For Eq (1), we also have ψ(x, t) = A 3 κ 1/ ( F (ξ) ± F (ξ) 1 + F (ξ) 1 c 0 + c F (ξ) c + c 4 F (ξ) 4) expiκx + Γx + Ω, (58) 4 ξ = A κx + A 1 A κ + A 4, Γ(t) = A 1 κ(t), 8 (A 1 + A c ) + 3 c4 A κ + A 5, 4 j=0 j=0 b(t) = c 4A κ t 8 κ, c 0 = 1

7 66 GE Jian-Ya, WANG Rui-Min, DAI Chao-Qing, and ZHANG Jie-Fang Vol 46 Therefore from Eq (58) and Table 1, equation (1) also has sn-type, cd-type, sc-type, sd-type solutions, and the corresponding soliton or solitonic solutions Although these solutions are only a small part of the large variety of possible solutions for Eq (1), they might serve as seeding solutions for a class of localized structures which exist in this system We hope that they will be useful in further perturbative and numerical analysis of various solutions to the nonlinear Schrödinger equations with variable-coefficient References 1 YuS Kivshar and BL Davies, Phys Rep 89 (1998) 81; N Akhmediev and A Ankiewicz, Solitons: Nonlinear Pulses and Beams, Chapman and Hall, London (1997); HA Haus and WS Wong, Rev Mod Phys 68 (1996) 43 A Hasegawa and F Tappet, Appl Phys Lett 3 (1973) 14 3 LF Mollenauer, RH Stolen, and JP Gordon, Phys Rev Lett 45 (1980) R Grimshaw, Proc Roy Soc London A 419 (1979) CS Cardner, JM Kruskal, and RM Miura, Phys Rev Lett 19 (1967) HD Wahlquist and FB Estabrook, Phys Lett 31 (1971) EG Fan, Phys Lett A 8 (001) 18; W Malfliet, Am J Phys 60 (199) SK Liu, et al, Phys Lett A 89 (001) 69; EG Parkes, BR Duffy, and PC Abbott, Phys Lett A 95 (00) 80 9 XY Tang, SY Lou, and Y Zhang, Phys Rev E 66 (00) ; SY Lou, Phys Lett A 77 (000) 94; JF Zhang, CL Zheng, JP Meng, and JP Fang, Chin Phys Lett 0 (003) Y Chen, B Li, and HQ Zhang, Commun Theor Phys (Beijing, China) 40 (003) ML Wang and YB Zhou, Phys Lett A 318 (003) 84 1 JF Zhang and P Han, Chin Phys Lett 1 (1994) 71; JF Zhang and FY Chen, Acta Phys Sin 50 (001) 1648 (in Chinese); HY Ruan and YX Chen, J Phys Soc Jpn 7 (003) 1350; HM Li and FM Wu, Chin Phys Lett 1 (004) 145; B Li and Y Chen, Chaos, Solitons and Fractals 1 (004) ML Wang, YM Wang, and JL Zhang, Chin Phys 1 (003) HN Xuan, CJ Wang, and DF Zhang, Z Naturforsch 59a (004) 196