Optical solitary wave solutions to nonlinear Schrödinger equation with cubic-quintic nonlinearity in non-kerr media

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1 MM Research Preprints MMRC AMSS Academia Sinica Beijing No. 22 December 2003 Optical solitary wave solutions to nonlinear Schrödinger equation with cubic-quintic nonlinearity in non-kerr media Zhenya Yan Key Laboratory of Mathematics Mechanization Institute of systems Science AMSS Chinese Academy of Sciences Beijing P.R. China Abstract. The study of optical solitary wave solutions is of prime significance for nonlinear Schrödinger equations with higher-order dispersions and/or higher-degree nonlinearities in non-kerr media. In this literature we investigate optical solitary wave solutions of nonlinear Schrödinger equation with cubic-quintic nonlinear terms and thirdorder dispersion which describes the effects of quintic nonlinearity on the ultrashort optical solitons pulse propagation in non-kerr media. Using the gauge transformation we reduce it to nonlinear ordinary differential equation. And then with the aid of symbolic computation and an unified transformation we show that it admits many types of optical solitary wave solutions in explicit analytic forms which include bright optical solitary wave solutions dark optical solitary wave solutions singular solitary wave solutions as well as the coupled bright-dark optical solitary wave solutions. Moveover we analysis their some properties. Keywords:Non-Kerr media; nonlinear Schrödinger equation; optical solitary wave solutions; symbolic computation 1. Introduction The study of optical solitary wave solutions is an interesting subject in soliton theory and telecommunication because of their capability of propagation long distance without attenuation and changing their shapes129]. When the pulse width is greater than 100 fs the celebrated nonlinear Schrödinger equation with cubic nonlinear terms is usually used to describe the propagation of light pulses in Kerr media. However when the intensity of the incident light filed gets stranger and stranger one can not neglect the non-kerr nonlinearity effects and nonlinear Schrodinger equations with higher order dispersion terms are needed to describe the propagation of optical pluses in fibers. For example the higher order nonlinear Schrödinger equation with third order dispersion self-steepening and self-frequency shift were studied3-5] which describes the propagation of femtosecond pulses in optical fibers. When high optical intensities are studied it is necessary to consider higher power nonlinearities. Recently Radhakrishnan et al.6] presented the following nonlinear Schrodinger equation with cubic-quintic nonlinearity ] i U z + 1vg U t 1 2 k ωωu tt i 6 k ωωωu ttt + kn U 2 U + kn 4β 0 U 4 U

2 Optical solitary wave solutions in v g ( U 2 U) t + in 4β 0 v g ( U 4 U) t = 0 (1) where the subscript ω denotes the differentiation of the wave number k with respect to ω the subscripts t z indicate the differentiation of U(z t) with respect to t z respectively v g is the group velocity is the linear refractive index coefficient n 2 n 4 are nonlinear refractive index coefficients and the parameters α 0 β 0 depend on the form of the function R(r) which describes the transverse field models( see 6] for details ). The model (1) describes the effects of quintic nonlinear terms. These terms in (1) contain the the group velocity dispersion self-phase modulation third order dispersion cubic terms and quintic terms. When the pulse widths become greater than 100 fs one can neglect the three terms i 6 k ωωωu ttt in v g ( U 2 U) t in 4β 0 v g ( U 4 U) t. Using dimensionless variables6] q = U A 0 t 1 z 0 ( k ωω ) (t z ) z z z 0 = v g 2z 0 kn 2 α 0 A 0 2 γ = 2n 4β 0 A 0 2 γ 1 = k ωωω 1 n 2 α 0 3k ωω z 0 k ωω γ 2 = 2 n A 0 2 γ 3 = 2n 4β 0 A 0 2 A 0 2 v g k k ωω v g k n 2 α 0 k ωω where A 0 is a measure of the maximum amplitude of the input pulse and z 0 characterizes the nonlinear properties of the fiber one can reduce (1) to the form iq z + q tt + 2 q 2 q + γ q 4 q + iγ 1 q ttt + iγ 2 ( q 2 q) t + iγ 3 ( q 4 q) t = 0 (2) In fact the parameters γ γ 1 γ 2 and γ 3 are not independent. For example from the above dimensionless variables we find that γ γ 2 and γ 3 have the relationship 2γ = γ 2 γ 3 (3) which is very important for us to seek optical solitary wave solutions of (1) below. There may exist other some relationships among these parameters γ γ 1 γ 2 and γ 3. Though (1) and (2) are equivalent under dimensionless variables it is difficult to find all possible relationships among these parameters γ γ 1 γ 2 and γ 3 which are useful to find solutions of (2). In general it is very different to find explicit exact solution of (1) or (2). Recently Hong7] used some transformations to obtain analytical bright and dark solitary wave solutions of (2). In this paper we will use our transformation10] to investigate directly (1) rather than (2) such that many new types of optical solitary wave solutions of (1) are obtained. The rest of this paper is organized as follows. In Sec. 2 we introduce a simply proposition which is useful to derive the optical solitary wave solutions of (1). In Sec. 3 we use the transformation presented in Sec. 2 to arrive at some optical solitary wave solutions of (1) which includes bright optical solitary wave solutions dark optical solitons singular solitons as well as the coupled bright-dark optical solitary wave solutions. Moreover it is easy to see that (1) is not Painleve integrable. Finally some conclusions and problems are given in Sec. 4.

3 344 Zhenya Yan 2. Introducing an useful proposition We know that many types of solitary wave solutions of nonlinear evolution equations consist of hyperbolic functions and hyperbolic functions consist of exponential functions. For instance tanh ξ = (e ξ e ξ )/(e ξ +e ξ ) sechξ = 2/(e ξ +e ξ ). We shall introduce an uniformed exponential function transformation 4A exp(ξ) (C + ) exp(2ξ) + 2( + 2B) exp(ξ) + C (4) which involve many types of solitary wave profiles (see Proposition in the following) where A 0 B C are parameters to seek solutions of (1). To express conveniently some obtained solutions of (1) below we firstly give the following property10]: Proposition: For the given transformation (4) we have the following equivalent forms using relationships between exponential functions and hyperbolic functions: Type 1: When 2B + 0 C + 0 Asech 2 ( 1 2 ξ) Bsech 2 ( 1 2 ξ) + C tanh( 1 (5) 2ξ) + Type 2: When B = > C 2 Type 3: When B = < C 2 ( 2A 2 C sech 2 ( 2A C 2 csch 2 Type 4: When ( + 2B) > 0 C = ± A tanh 1 ( + 2B 2 Type 5: When ( + 2B) < 0 C = ± A coth 1 ( + 2B 2 ξ 1 2 ln C + C ξ 1 2 ln C + C ξ ln + 2B ξ ln + 2B ) (6) ) (7) ) ] + 1 (8) ) ] + 1 (9) Remark 1: When A = 1 = 4 C = 0 µ = 1 the transformation (4) reduces to another one due to Zhang and Ma8]. But our more general form (4) can be used to seek more types of solutions of some nonlinear differential equations. 3. Optical solitary wave solutions for equation (1)

4 Optical solitary wave solutions 345 In the following we shall use the above-mentioned transformation (4) to seek optical solitary wave solutions of (1). First of all we make the gauge transformation in the form U(z t) = u(ξ) exp(iη) ξ = ρ(t + λz) + c η = αt + βz (10) where c is an arbitrary constant ρ λ α β are real parameters ρ/λ 1/λ α β are the amplitude the velocity frequency shifts and wavenumber respectively and u(ξ) is a real function of ξ). The substitution of (10) into (1) yields iρλu + ρ v g u k ωω ραu k ωωω 6 (ρ3 u 3α 2 ρu ) + 3n v g ρu 2 u + 5n 4β 0 v g ρu 4 u ] βu α v g u 1 2 k ωω(ρ 2 u α 2 u) k ωωω 6 ( 3αρ2 u + α 3 u) + kn u 3 + kn 4β 0 u 5 n v g αu 3 n 4β 0 v g αu 5 = 0 (11) whose real and imaginary parts give rise to the system of nonlinear differential equations k ωωω 6 ρ2 u + (λ + 1 v g k ωω α k ωωωα 2 )u + 3n v g u 2 u + 5n 4β 0 v g u 4 u = 0 (12a) ( 1 2 k ωωρ k ωωωαρ 2 )u + ( β α v g k ωωα k ωωωα 3 )u +( kn n v g α)u 3 + ( kn 4β 0 n 4β 0 v g α)u 5 = 0 (12b) respectively where the prime denotes the differentiation of u with respect to ξ. By integrating (12a) with respect to ξ once and setting the integration constant to zero we get k ωωω 6 ρ2 u + (λ + 1 v g k ωω α k ωωωα 2 )u + n v g u 3 + n 4β 0 v g u 5 = 0 (13) It is easy to see from (12b) and (13) that the same real function u(ξ) is required to satisfy two equations (12b) and (13) simultaneously. Therefore the ratios of their corresponding terms should be equal that is k ωωω k ωω k ωωωα = λ + 1 v g k ωω α k ωωωα 2 β α v g k ωωα k ωωωα 3 = n 2 α 0 v g kn 2 α 0 n α v g = n 4 β 0 v g kn 4 β 0 n (14) 4β 0 α v g

5 346 Zhenya Yan from which we gain α = 3k ωω 2k ωωω 1 2 kv g β = α v g k ωωα k ωωωα 3 +(α kv g )(λ + 1 v g k ωω α k ωωωα 2 ). (15) Therefore under the condition (15) (12b) and (13) are equivalent. In the following we only consider nonlinear ordinary differential equation (13). If we further make the transformation u 2 (ξ) = w(ξ) then (13) reduces to k ωωω 6 ρ2 ( 1 4 w ww ) + (λ + 1 v g k ωω α k ωωωα 2 )w 2 + n v g w 3 + n 4β 0 v g w 4 = 0 (16) Assume that (16) has the solution in the form w(ξ) = 4Ae ξ (C + )e 2ξ + 2( + 2B)e ξ + F (17a) + C where A B C F are parameters to be determined later. With the aid of symbolic computation we derive from (17a) that w = 4Ae ξ (C + )e 2ξ + 2( + 2B)e ξ + C 4Aeξ 2(C + )e 2ξ + 2( + 2B)e ξ ] (C + )e 2ξ + 2( + 2B)e ξ + C] 2 (17b) w = 8Aeξ 2(C + )e 2ξ + 2( + 2B)e ξ ] 2 (C + )e 2ξ + 2( + 2B)e ξ + C] 3 + 4Ae ξ (C + )e 2ξ + 2( + 2B)e ξ + C Aeξ 32(C + )e 2ξ + 24( + 2B)e ξ ] (C + )e 2ξ + 2( + 2B)e ξ + C] 2 (17c) We substitute (17a)-(17c) into the left-hand side of (16) to get an expression with respect to exp(ξ) and collect the numerator of the expression to gain a polynomial in exp(ξ). Because these terms exp(iξ)(i = ) are linearly independent thus setting to zero their coefficients gives rise to a system of algebraic equations with respect to the unknowns A B C F ρ λ. (Because the system is so complicated we omit it here.) With the aid of symbolic computation by solving the system we obtain the following possible nontrivial solutions: Case 1: When B 1 2 C ± A = 3n (C 2 + 4B + 4B 2 ) ρ 2 = 9n2 2 α2 0 (C2 + 4B + 4B 2 ) ( + 2B) 2k ωωω v g n 4 β 0 (µ + 2B) 2 F = 0 λ = 1 v g + k ωω α 1 2 k ωωωα 2 + 3n2 2 α2 0 (C2 + 4B + 4B 2 ) 16v g n 4 β 0 ( + 2B) 2. (18)

6 Optical solitary wave solutions 347 Case 2: When B = 1 2 µ 2 > C 2 λ = 1 v g + k ωω α 1 2 k ωωωα n2 2 α2 0 64v g n 4 β 0 ρ 2 9n 2 2 = α2 0 F = 3n A = 3n 2 C 2. (19) 8k ωωω v g n 4 β 0 16n 4 β 0 Case 3: When C = ± F = 0 λ = 1 v g + k ωω α 1 2 k ωωωα 2 + 3n2 2 α2 0 16v g n 4 β 0 ρ 2 = Case 4: When C = ± ρ 2 = 9n 2 2 α2 0 A = 3n (µ + 2B). (20) 2k ωωω v g n 4 β 0 λ = 1 v g + k ωω α 1 2 k ωωωα 2 + 3n2 2 α2 0 16v g n 4 β 0 9n 2 2 α2 0 2k ωωω v g n 4 β 0 F = 3n 4n 4 β 0 A = 3n (µ + 2B). (21) Therefore we can arrive at the following optical solitary wave solutions of (1) in terms of the proposition in the second section Case 1-4 (10) (17a) and transformation u 2 = w. According to Case 1 and Type 1-2 we have the optical solitary wave solutions of (1): Family 1: The coupled bright-dark optical solitary wave solution with µ = 1 U 1 = 3n (C 2 + 4B + 4B 2 ) ( + 2B) sech ξ Bsech ξ + C tanh 1 2 ξ + ] 1/2 expi(αt + βz)] (22) where B 1 2 C ± ξ = ρ(t + λz) + c λ and ρ are defined by (18) and α β are given by (15). Because B C are three arbitrary constants thus the properly chosen parameters B C can guarantee Bsech ξ + C tanh 1 2ξ + 0 for any ξ. For example if we take B > 0 > 0 > C then Bsech ξ + C tanh 1 2ξ + > 0 for any ξ. When ξ U 1 tends to zero while when ξ 0 U 1 tends to the constant 3n2 α 0 (C 2 + 4B + 4B 2 )/ ( + 2B)(B + )]. Thus this solution (22) is a regular optical solitary wave solution which consists of bright and dark optical solitary wave solutions. But for other parameters values of B C there may be the singular point z = z 0 for the fixed time t = t 0. For example when C = 0 B = = 1 U 1 will blow up at the point z = z 0 for the fixed time t = t 0 where z 0 satisfies 1 2 ρ(t 0 + λz 0 ) + c = ±1 ± 2. In particular when C = 0 B > 0 we have the bright optical solitary wave solutions of (1) U 11 = 3n (4B + 4B 2 ) sech ξ ] 1/2 ( + 2B) Bsech ξ + expi(αt + βz)] (23)

7 348 Zhenya Yan When ξ U 1 approaches to 0 while when ξ 0 U 1 approaches to a constant i.e. 3n2 α 0 (4B + 4B 2 )/ ( + 2B)(B + )]. According to Case 2 and Type 2 we have the bright optical solitary wave solutions of (1): Family 2: The bright optical solitary wave solutions U 2 = ( 3n2 α 0 sech ξ 1 2 ln C + C ) 3n ] 1/2 expi(αt + βz)] (24) where n 2 n 4 α 0 β 0 ) < 0 ξ = ρ(t+λz)+c and λ ρ are defined by (19). When ξ 1 2 U 2 approaches to (3n 2 α 0 )/( ) while when ξ 1 2 ln C +C ln C +C 0 U 2 approaches to 0. According to Case 3 and Type 4-5 we have optical solitary wave solutions of (1): Family 3: The dark optical solitary wave solution U 3 = 3n tanh 1 + 2B (ξ ln 2 ) 3n ] 1/2 expi(αt + βz)] (25) where n 2 α 0 n 4 β 0 < 0 ξ = ρ(t + λz) + c and λ ρ are defined by (20) and ( + 2B) > 0. We know that when ξ + ln +2B ± U 2 approaches to (3n 2 α 0 )/(4n 4 β 0 ) and 0 while U 2 approaches to (3n 2 α 0 )/( ) at the hole center ξ ln +2B = 0. Family 4: The singular optical solitary wave solution U 4 = 3n coth 1 + 2B (ξ ln 2 ) 3n ] 1/2 expi(αt + βz)] (26) where ξ = ρ(t + λz) + c and λ ρ are defined by (20) and (µ + 2B) < 0. There exists the point z = z 0 for the fixed time t = t 0 at which the solution will below up. According to Case 4 and Type 4-5 we have the optical solitary wave solutions of (1): Family 5: The dark optical solitary wave solution U 5 = 3n2 α 0 tanh 1 + 2B (ξ + ln 2 ) 3n ] 1/2 expi(αt + βz)] (27) where n 2 α 0 n 4 β 0 < 0 ξ = ρ(t + λz) + c and λ ρ are defined by (21) and ( + 2B) > 0. We know that when ξ + ln +2B ± U 2 approaches to 0 and (3n 2 α 0 )/(4n 4 β 0 ) while U 2 approaches to (3n 2 α 0 )/( ) at the hole center ξ + ln +2B = 0. Family 6: The singular optical solitary wave solution U 6 = 3n2 α 0 coth 1 + 2B (ξ + ln 2 ) 3n ] 1/2 expi(αt + βz)] (28) where ξ = ρ(t + λz) + c λ ρ are defined by (21) and ( + 2B) < 0. There exists the point z = z 0 for the fixed time t = t 0 at which the solution will below up.

8 Optical solitary wave solutions 349 Remark 2: The bright and dark optical solitary wave solutions of (1) obtained here are different from ones derived by Hong7]. In addition we also gain other new types of optical solitary wave solutions of (1). These solutions may be useful to explain some physical phenomena and to further consider the wave propagation in optical fibres. Remark 3: We know that (13) is one ordinary differential equation reduction of (1). We now consider the Painlevé integrability of (13). Let11] The substitution of (29) into (13) leads to u u 0 (ξ ξ 0 ) p. (29) p = 1 2 u4 0 = k ωωωρ 2 v g. (30) Thus it can be proved that (13) has a movable branch point of order 1 2 (13) is not Painlevé integrable. Therefore (1) has no Painlevé property. which denotes that 4. Conclusions and remarks In summary we reduce (1) to nonlinear ordinary differential equation (13) using the gauge transformation (10) with (15). And then by using w = u 2 and the formal solution transformation (17a) we derive many types of optical solitary wave solutions of (1) which include the bright optical solitary wave solutions the dark optical solitary wave solutions the coupled bright-dark optical solitary wave solutions and singular optical solitary wave solutions. Moreover we analysis some properties of these obtained solutions. These solutions may be useful to explain propagation rules of pulses in non-kerr media. We also prove that (1) is not Painleve integrable using the Painleve integrability of the reduction equation (13). A natural problem is whether there exist other types of exact solutions for (1) such as elliptic function solutions. Further study is needed in future. References 1] G. P. Agarwal Nonlinear Fiber Optic 2nd ed. Accademic New York ] A. Hasegawa and K. Kodama Solitons in Optical Communications Oxford University Press New York ] M. Gedalin et al. Phys. Rev. Lett. 78(1997) ] S. L. Palacios et al. Phys. Rev. 60(1999) R45. 5] Z. Y. Yan Chaos Solitons and Fractals 16(2003) ] R. Radhakrishnan et al. Phys. Rev. 60(1999) ] W. P. Hong Opt. Commun. 194(2001) ] W. G. Zhang and W. X. Ma Appl. Math. Mech. 20(1999) ] Y. S. Kivshar and B. Luther-Davies Phys. Rep. 298(1998) ] Z. Y. Yan and H. Q. Zhang Proc. of 5th Asia Symp. Comput. Math. World Science Singapore 2001 p ] M.J. Ablowitz et al. J. Math. Phys. 21(1980) 715.