Nonautonomous solitons and interactions for a variable-coefficient resonant nonlinear Schrödinger equation

Applied Mathematics Letters 60 (2016) 8–13

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Applied Mathematics Letters www.elsevier.com/locate/aml

Nonautonomous solitons and interactions for a variable-coefficient resonant nonlinear Schr¨odinger equation Min Li a,∗ , Tao Xu b , Lei Wang a , Feng-Hua Qi c a

Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China College of Science, China University of Petroleum, Beijing 102249, China c School of Information, Beijing Wuzi University, Beijing 101149, China b

article

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Article history: Received 28 February 2016 Received in revised form 23 March 2016 Accepted 23 March 2016 Available online 5 April 2016 Keywords: Variable-coefficient resonant nonlinear Schr¨ odinger equation Resonant interactions Intermediate-state soliton interactions Binary Bell polynomials

abstract A variable-coefficient resonant nonlinear Schr¨ odinger (vc-RNLS) equation is considered in this paper. Binary Bell polynomials are employed to obtain the bilinear form and multi-soliton solutions under the integrable conditions. Four types of nonautonomous solitons are derived including the parabolic soliton, compressed soliton, phase-shifted soliton and periodic soliton. Propagation dynamics for each type is analyzed in detail. Nonautonomous resonant and intermediate-state soliton interactions are found to be existent under certain conditions. Specially, periodic soliton interactions are discussed, which shows that the periodic dispersion has no effect on the generation of resonance and intermediate-state solitons. Those analysis might have the applications in optical communication systems with the black hole physics flavor. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Nonlinear evolution equations arising in various areas have been intensively studied in the past few years [1]. Of those, nonlinear Schr¨ odinger (NLS) equation is a universal equation of the nonlinear integrable systems [2]. Due to the applications in nonlinear optics and plasma physics, the soliton solutions of NLS equation have been intensively studied via kinds of mathematical methods, such as the inverse scattering transform, Darboux transformation, and Hirota bilinear method [3]. The relative sign of dispersion and nonlinear terms divides the NLS equation into focusing and defocusing types [2]. On the other hand, the quadratic dispersion can be seen as the combination of the phase and modulus dispersions, which in general have the same sign [4]. But the sign may be opposite in the phenomenological description of the hypothetical nonlinear media [5]. Thus, a novel intermediate equation between the focusing and defocusing NLS equations ∗ Corresponding author. E-mail address: [email protected] (M. Li).

http://dx.doi.org/10.1016/j.aml.2016.03.014 0893-9659/© 2016 Elsevier Ltd. All rights reserved.

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has been proposed, i.e., the resonant NLS (RNLS) equation [4]: iut + uxx +

α 2 1 |u| u = β |u|xx u, 4 |u|

(1)

where u = u(x, t) denotes the complex amplitude, α is the nonlinear coefficient, and β denotes the strength of electrostriction pressure or diffraction. Eq. (1) can describe the response of a hypothetical resonance medium to an action of a quasimonochromatic wave [4]. The term |u|xx /|u| represents the “quantum potential”. Eq. (1) with β > 1 can support the resonant interactions [6,7], which have potential applications in optical communication systems with the black hole physics flavor [8]. Considering the inhomogeneity of the resonance medium, in this paper, we investigate the following variable-coefficient RNLS (vc-RNLS) equation [9], iUt + α(t)Uxx + β(t)|U |2 U + γ(t)

1 |U |xx U = 0, |U |

(2)

where U = U (x, t) represents the complex wave profile, x and t respectively denote the spatial and temporal variables, α(t) and β(t) are the dispersion and nonlinear coefficients, γ(t) is the resonant term and can be referred to as the inhomogeneous quantum potential. In Section 2, we will construct the bilinear form of Eq. (2) via binary Bell polynomials. Such method can avoid the complexity induced by the resonant term. Soliton solutions will be derived by means of the parameter expansion method. Based on the exact solutions, in Section 3, we will analyze the nonautonomous soliton interactions and give the conditions for the generation of resonant and intermediate-state soliton interactions. Moreover, the dynamics of periodic soliton interactions will be discussed in detail. Section 4 will be our conclusions. 2. Soliton solutions via binary Bell polynomials 2.1. Bilinear form First of all, we derive the bilinear form of Eq. (2) by means of the binary Bell polynomials. With the transformation, U = exp

u + iθu + p − iθp , 2

(3)

where u = u(x, t) and p = p(x, t) are both real functions, and θ is a real constant, Eq. (2) is converted into the following form without any denominator, 1 (i − θ)ut + [α(t) − θ2 α(t) + γ(t) + 2i θ α(t)] u2x + [α(t) + γ(t) + i θ α(t)] uxx 2 1 + (i + θ) pt + [α(t) − θ2 α(t) + γ(t) − 2i θ α(t)] p2x + [α(t) + γ(t) − i θ α(t)] pxx 2 + [α(t) + γ(t) + θ2 α(t)] ux px + 2β(t) exp(u + p) = 0.

(4)

Further, we introduce two C ∞ auxiliary functions v(x, t) and q(x, t) to form two sets of binary Bell polynomials with u(x, t) and p(x, t), respectively. Under the integrable conditions γ(t) = −(1 + θ2 )α(t) and β(t) = η α(t) with η being an arbitrary constant, the linear binary-Bell-polynomial form of Eq. (2) is derived as follows: Yt (u, v) + θ α(t)Y2x (u, v) = 0, 2

θ P2x (q − p) + η exp(u + p) = 0,

Yt (p, q) − θ α(t)Y2x (p, q) = 0,

(5) (6)

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where P2x is the P -polynomials defined in Refs. [10] and two sets of binary Bell polynomials are involved by virtue of two auxiliary functions. Via the dependent variable transformations u = ln

g , f

v = ln(g f ),

p = ln

h , f

q = ln(h f ),

(7)

with g = g(x, t) and f = f (x, t), the bilinear form of Eq. (2) is given as below, (Dt + θ α(t)Dx2 )(g · f ) = 0,

(Dt − θ α(t)Dx2 )(h · f ) = 0,

θ2 Dx2 (f · f ) + η g h = 0.

(8)

By expanding g, h and f with respect to a formal expansion parameter ε as g = εg1 + ε3 g3 + · · · ,

h = εh1 + ε3 h3 + · · · ,

f = 1 + ε2 f2 + ε4 f4 + · · · ,

(9)

where gl = gl (x, t), hl = hl (x, t) (l = 1, 3, 5 · · ·) and fm = fm (x, t) (m = 2, 4, 6 · · ·) are all real functions to be determined, we can obtain the multi-soliton solutions of Eq. (2). 2.2. Soliton solutions Truncating the expansions (9) as g = εg1 , h = εh1 and f = 1 + ε2 f2 , and substituting them into the bilinear form (8), we derive the one-soliton solution as 1+iθ 2

U = g1

1−iθ 2

h1

/(1 + f2 )

(10)

with  2 φj1 = kj1 x + wj1 (t), w11 (t) = −θ k11 α(t)dt − ln c11 (t),  2 w21 (t) = θ k21 α(t)dt − ln c21 (t), g1 = c11 (t) exp(φ11 ) (j = 1, 2), h1 = c21 (t) exp(φ21 ),

f2 = χ1 (t) exp(φ11 + φ21 ),

χ1 (t) = −

κ c11 (t)c21 (t) , 2θ2 (k11 + k21 )2

where kj1 and cj1 (t) (j = 1, 2) are arbitrary constants and functions, respectively. Similarly, by substituting the truncated terms g = εg1 + ε3 g3 , h = εh1 + ε3 h3 and f = 1 + ε2 f2 + ε4 f4 into the bilinear form (8), we obtain the two-soliton solution of Eq. (2), U = (g1 + g3 )

1+iθ 2

(h1 + h3 )

1−iθ 2

/(1 + f2 + f4 )

(11)

with  2 φjl = kjl x + wjl (t), w1l (t) = −θ k1l α(t)dt − ln c1l (t) (j = 1, 2, l = 1, 2),  η cm1 (t)cm2 (t)co(n−2) (t)(km1 − km2 )2 2 w2l (t) = θ k2l α(t)dt − ln c2l (t), cmn (t) = − 2 , 2θ (km1 + ko(n−2) )2 (km2 + ko(n−2) )2 f4 = χ5 (t) exp(φ11 + φ12 + φ21 + φ22 ),

χpq (t) = −

η c1p (t)c2q (t) , 2 θ2 (k1p + k2q )2

η 2 c11 (t)c12 (t)c21 (t)c22 (t)(k11 − k12 )2 (k21 − k22 )2 , 64 s4 (k11 + k21 )2 (k12 + k21 )2 (k11 + k22 )2 (k12 + k22 )2 g1 = c11 (t) exp(φ11 ) + c12 (t) exp(φ12 ), h1 = c21 (t) exp(φ21 ) + c22 (t) exp(φ22 ), χ5 (t) =

g3 = c13 (t) exp(φ11 + φ12 + φ21 ) + c14 (t) exp(φ11 + φ12 + φ22 ), h3 = c23 (t) exp(φ21 + φ22 + φ11 ) + c24 (t) exp(φ21 + φ22 + φ12 ), f2 = χ11 (t) exp(φ11 + φ21 ) + χ21 (t) exp(φ12 + φ21 ) + χ12 (t) exp(φ11 + φ22 ) + χ22 (t) exp(φ12 + φ22 ), where m = 1, 2, n = 3, 4, o = 1 or 2 if m = 2 or 1, p = 1, 2, q = 1, 2. The nonsingularity of the solution (11) requires η < 0, which is consistent with that in Refs. [6,11].

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2

1 8

1 0 t

-5 x 2

-5

3

3 |U| 0

|U|

|U| 0

x

4

0

15

-15

-5

t

-6

t x

-3

t

x

-15

5

15

-10

Fig. 1. (a) Parabolic soliton via the solution (10) with θ = 0.5, η = −1, k11 = 1, k21 = 1.5 and α(t) = t − 1; (b) Soliton pulse compression via the solution (10) with θ = 1, η = −1, k11 = 1, k21 = 2 and α(t) = e0.4t ; (c) Soliton phase shift via the solution (10) with θ = 1, η = −1, k11 = 0.1, k21 = 0.5 and α(t) = 1 − 10 sech(t); (d) Periodic soliton via the solution (10) with θ = 1, η = −1, k11 = 1, k21 = 2 and α(t) = sin(t).

3. Nonautonomous solitons and interactions 3.1. Propagations of the nonautonomous solitons As to the one-soliton solution (10), the intensity of the single soliton can be described by 2

2

4θ4 (k11 + k21 )4 e(k11 +k21 )x+∆(k11 +k21 ) |U (x, t)| = 2 2 2 2θ (k11 + k21 )2 e∆ k11 − η e(k11 +k21 )x+∆ k21 2

 with ∆ = θ

α(t)dt.

(12)

Although there are three arbitrary functions in the solution (10), only the arbitrary function α(t) can influence the intensity of the soliton. In the following, we will choose several types of function α(t) to analyze the soliton dynamics in inhomogeneous systems. Parabolic soliton—If we choose the dispersion parameter α(t) in the form of linear function about the variable t, i.e., α(t) = ρ1 t + ρ2 , where ρ1 and ρ2 are two arbitrary constants, the parabolic soliton can be obtained. ρ1 determines the open direction of the parabola and ρ2 influences the position of the soliton along the t-axis. As an example, by choosing α(t) = t − 1 in Fig. 1(a), after propagating a certain distance the parabolic soliton will reverse and evolve along the opposite direction. Besides, the pulse experiences the process of broadening and compression. Such kind of soliton has also been derived in an inhomogeneous NLS equation with frequency chirping and named as Boomerang soliton [12]. Soliton pulse compression—As we know, the pulse compression in optical communications is important. Hereby, we consider the function α(t) in the exponential form, i.e., α(t) = eµ t , where µ is an arbitrary constant. Because the coefficients of dispersion and nonlinear are both proportional to the function α(t), the soliton may not be compressed exponentially during the propagation even though we choose the decreasing dispersion with µ < 0. On the contrary, µ > 0 leads to the pulse compression, while the soliton will get broadened when µ < 0. Fig. 1(b) shows the case of µ = 0.4, where the soliton not only gets compressed but also changes its direction. Soliton phase shift—With the dispersion function chosen as α(t) = ϱ1 sech(t) + ϱ2 , large soliton phase shift will arise during the propagation. Moreover, it is found that the sign and absolute value of ϱ1 determines the direction and distance of the phase shift, respectively. The parameter ϱ2 affects the velocity of the soliton. By setting α(t) = −10 sech(t) + 1, the soliton experiences a large phase shift around t = 0 but has no change in the shape, as seen in Fig. 1(c). Periodic soliton—In order to obtain the periodic distributed system, we choose the dispersion function α(t) in the form α(t) = R1 sin(R2 t), where R1 and R2 affect the amplitude and frequency of the periodic trajectory, respectively. Fig. 1(d) shows the propagation of a soliton with periodic oscillation along the distance. The inhomogeneities only lead to the periodic oscillation in space and time, but have no effect on the shape of the soliton.

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3

2

3 6

0 t

-10 x

20

-6

|U|

20 0 t

-15

|U|

3 10

0

|U|

-5 t x

x

15

-20

30

-10

0 -40

t x

40

-30

Fig. 2. (a) Periodic resonant interaction via the solution (11) with θ = 1, η = −1, k11 = k12 = 0.8, k21 = 0.5, k22 = 0.9 and α(t) = 10 sin(t); (b) Periodic head-on interaction generating large-amplitude intermediate soliton via the solution (11) with θ = 1, η = −1, k11 = 1, k12 = 0.5, k21 = 0.8, k22 = −0.499 and α(t) = 5 sin(0.2t); (c) Periodic head-on interaction generating smallamplitude intermediate soliton via the solution (11) with θ = 1, η = −1, k11 = 0.3, k12 = 0.8, k21 = 0.59999, k22 = 0.6 and α(t) = 5 sin(t); (d) Periodic overtaking interaction generating small-amplitude intermediate soliton via the solution (11) with θ = 1, η = −1, k11 = 1, k12 = 0.5, k21 = −0.999, k22 = 0.8, and α(t) = 3 sin(0.15t).

3.2. Nonautonomous soliton resonance and intermediate-state soliton interactions For Eq. (1), there exist rich nonlinear phenomena such as resonant and intermediate-state soliton interactions [7]. Note that, the types of the soliton interactions for Eq. (2) still rely on the parameter χ5 (t) in the solution (11). The condition χ5 (t) = 0 or ∞, i.e., (k11 − k12 )(k21 − k22 ) = 0,

or

(k11 + k21 )(k12 + k21 )(k11 + k22 )(k12 + k22 ) = 0,

(13)

leads to the resonant interactions of two solitons, while χ5 (t) → ∞ or → 0, i.e., (k11 − k12 )(k21 − k22 ) → 0,

or

(k11 + k21 )(k12 + k21 )(k11 + k22 )(k12 + k22 ) → 0,

(14)

will generate the large- or small-amplitude intermediate solitons during the interactions. Considering the complexity of interactions and potential applications of periodic compression/ amplification, we only analyze the case of periodic function, as seen in Fig. 2. By choosing k11 = k12 and α(t) = 10 sin(t), the periodic resonant interaction arises in Fig. 2(a). Two small-amplitude solitons interact with each other and fuse into one large-amplitude soliton. Such process will repeat periodically. Fig. 2(b) displays the periodic head-on interaction generating large-amplitude intermediate soliton with k12 ∼ −k22 ≪ k21 ≪ k11 and α(t) = 5 sin(0.2t). Different from the fusion, the large-amplitude intermediate soliton further splits up into two small-amplitude ones with original velocities. Because the velocity of the intermediate soliton is not zero, the trajectory presents a periodic polygonal line. With k11 ≪ k21 ∼ k22 ≪ k12 and α(t) = 5 sin(t), the periodic head-on interaction generating small-amplitude intermediate soliton is obtained in Fig. 2(c). The left part of the figure depicts the splitting process of one soliton in one period. Then the split intermediate soliton eventually fuses with the right-side soliton. By adopting k11 ∼ −k21 ≫ k12 ≫ k22 and α(t) = 3 sin(0.15t), we get the periodic overtaking interaction generating small-amplitude intermediate soliton in Fig. 2(d). During one period, one large-amplitude soliton splits up into one intermediate soliton and one small-amplitude soliton. Further, the intermediate one fuses with another small-amplitude soliton into the original soliton. Based on the analysis, it is found that the periodic function only leads to soliton interacting periodically but has no effect on the properties of the interactions. 4. Conclusions In this paper, we have considered the vc-RNLS equation (i.e., Eq. (2)). Bilinear form and soliton solutions of Eq. (2) under the integrable conditions have been derived via the binary Bell polynomials. By choosing different dispersion function α(t), we have obtained four types of nonautonomous solitons, i.e., the parabolic soliton, compressed soliton, phase-shifted soliton and periodic soliton (see Fig. 1). Propagation dynamics for

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each type has been analyzed in detail. It is found that resonant and intermediate-state soliton interactions also exist in the inhomogeneous media under the conditions (13) and (14), respectively. Specially, periodic soliton interactions have been analyzed in Fig. 2, which shows that the periodic dispersion function only leads to soliton interacting periodically but has no effect on the generation of the resonance and intermediate-state solitons. Our analysis might have the applications in optical communication systems with the black hole physics flavor. Acknowledgments This work has been supported by the Fundamental Research Funds of the Central Universities (Project Nos. 2014QN30, 2014ZZD10 and 2015ZD16), by the National Natural Science Foundations of China (Grant Nos. 61505054, 11426105, 11305060, 11371371, 11271266 and 11271126), by the Natural Science Foundation of Beijing, China (Grant No. 1162003), and by the Higher-Level Item Cultivation Project of Beijing Wuzi University (No. GJB20141001). References [1] D.S. Wang, X.Q. Wei, Appl. Math. Lett. 51 (2016) 60; D.S. Wang, Y.B. Yin, Comput. Math. Appl. 71 (2016) 748; D.S. Wang, X.H. Hu, J.P. Hu, W.M. Liu, Phys. Rev. A 81 (2010) 025604. [2] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic, San Diego, 2001; Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic, San Diego, 2003. [3] G.D. Lyng, P.D. Miller, Comm. Pure Appl. Math. 60 (2007) 951; T.C. Xia, X.H. Chen, D.Y. Chen, Chaos Solitons Fractals 26 (2005) 889; J.R. Yan, L.X. Pan, J. Lu, Chin. Phys. 13 (2004) 441. [4] O.K. Pashaev, J.H. Lee, Modern Phys. Lett. A 17 (2002) 1601. [5] J.H. Lee, O, K. Pashaev, C. Rogers, W.K. Schief, J. Plasma Phys. 73 (2007) 257. [6] J.H. Lee, O.K. Pashaev, Theoret. Math. Phys. 152 (2007) 991; O.K. Pashaev, J.H. Lee, C. Rogers, J. Phys. A 41 (2008) 452001. [7] M. Li, J.H. Xiao, T.Z. Yan, B. Tian, Nonlinear Anal.-Real 14 (2013) 1669. [8] O.K. Pashaev, J.H. Lee, Modern Phys. Lett. A 17 (2002) 1601. [9] H. Triki, T. Hayat, O.M. Aldossary, A. Biswas, Opt. Laser Technol. 44 (2012) 2223. [10] F. Lambert, J. Springael, Acta Appl. Math. 102 (2008) 147. [11] O.K. Pashaev, J.H. Lee, ANZIAM J. 44 (2002) 73. [12] B. Li, Y. Chen, Chaos Solitons Fractals 33 (2007) 532.