Cross-modulation coupling of incoherent soliton modes in photorefractive crystals

Kutuzov, V.; Petnikova, V M; Shuvalov, Vladimir V; Vysloukh, Victor A.

doi:10.1103/physreve.57.6056

Cited by 63 publications

(32 citation statements)

References 46 publications

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“…In particular, we point out that the shape changing inelastic collision property persists for the N ≥ 3 cases also as in the N = 2 (Manakov) case reported recently [5], giving rise to many possibilities of energy exchange. Furthermore, we point out that in the context of spatial solitons the partially coherent stationary solitons(PCS) reported in the recent literature [9][10] are special cases of the above general soliton solutions which undergo shape changing collisions.The bright one-soliton and two-soliton solutions of the 3-CNLS system, iq jz + q jtt + 2µ(|q 1 | 2 + |q 2 | 2 + |q 3 | 2 )q j = 0, j = 1, 2, 3, (2) can be obtained from its equivalent bilinear form resulting from the transformation q j = g (j) /f , (iD z + D 2 t )g (j) .f = 0, D 2 t f.f = 2µwhere * denotes the complex conjugate, D z and D t are Hirota's bilinear operators [11], and g (j) 's are complex functions, while f (z, t) is a real function.The resulting set of Eqs. (3) can be solved recursively by making the power series expansion g (j)…”

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Exact Soliton Solutions, Shape Changing Collisions, and Partially Coherent Solitons in Coupled Nonlinear Schrödinger Equations

2001

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We present the exact bright one-soliton and two-soliton solutions of the integrable three coupled nonlinear Schrödinger equations (3-CNLS) by using the Hirota method, and then obtain them for the general N -coupled nonlinear Schrödinger equations (N -CNLS). It is pointed out that the underlying solitons undergo inelastic (shape changing) collisions due to intensity redistribution among the modes. We also analyse the various possibilities and conditions for such collisions to occur. Further, we report the significant fact that the various partial coherent solitons (PCS) discussed in the literature are special cases of the higher order bright soliton solutions of the N -CNLS equations.PACS numbers: 42.81. Dp, 42.65.Tg In recent years the concept of soliton has been receiving considerable attention in optical communications since soliton is capable of propagating over long distances without change of shape and with negligible attenuation [1][2][3]. It has been found that soliton propagation through optical fiber arrays is governed by a set of equations related to the CNLS equations [1,2],where q j is the envelope in the jth core, z and t represent the normalized distance along the fiber and the retarded time, respectively. Here 2µ gives the strength of the nonlinearity. Eq. (1) reduces to the standard envelope soliton possessing integrable nonlinear Schrödinger equation for N = 1. For N = 2, the above Eq. (1) governs the integrable Manakov system [4] and recently for this case the exact two-soliton solution has been obtained and novel shape changing inelastic collision property has been brought out [5]. However, the results are scarce for N ≥ 3, even though the underlying systems are of considerable physical interest. For example, in addition to optical communication, in the context of biophysics the case N = 3 can be used to study the launching and propagation of solitons along the three spines of an alpha-helix in protein [6]. Similarly the CNLS Eq.(1) and its generalizations for N ≥3 are of physical relevance in the theory of soliton wavelength division multiplexing [7], multi-channel bit-parallel-wavelength optical fiber networks [8] and so on. In particular, for arbitrary N , Eq.(1) governs the propagation of N -self trapped mutually incoherent wavepackets in Kerr-like photorefractive media [9] in which q j is the jth component of the beam, z and t represents the normalized coordinates along the direction of propagation and the transverse coordinate, respectively, and N p=1 |q p | 2 represents the change in the refractive index profile created by all incoherent components of the light beam [9] when the medium response is slow.The parameter µ = k 2 0 n 2 /2, where n 2 is the nonlinear Kerr coefficient and k 0 is the free space wave vector.In this letter, we report the exact bright one and two soliton solutions, first for the N = 3 case and then for the arbitrary N case, where the procedure can be extended in principle to higher order soliton solutions, using the Hirota bilinearization method. In particular, ...

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Exact Soliton Solutions, Shape Changing Collisions, and Partially Coherent Solitons in Coupled Nonlinear Schrödinger Equations

2001

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“…These equations are important for a number of physical applications. For example, for photorefractive media with a drift mechanism of nonlinear response, a good approximation describing the propagation of n self-trapped mutually incoherent wave packets is the set of equations for a Kerr-type nonlinearity [29] i ∂ ∂z Inserting (1.3) in (1.2) and renormalising the variables asQ j = Q j / √ 2α, z = 2t, x = x we obtain the vector nonlinear Schrödinger equation (1.1). Stability, localization, and soliton asymptotics of multicomponent photorefractive cnoidal waves are discussed in [35].…”

Section: Introductionmentioning

confidence: 99%

Quasiperiodic and periodic solutions for vector nonlinear Schrödinger equations

Eilbeck¹,

Enolskii²,

Kostov

2000

Journal of Mathematical Physics

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Abstract. We consider quasi-periodic and periodic (cnoidal) wave solutions of a set of n-component vector nonlinear Schrödinger equations (VNLSE). In a biased photorefractive crystal with a drift mechanism of nonlinear response and Kerr-type nonlinearity, n component nonlinear Schrödinger equations can be used to model self-trapped mutually incoherent wave packets. These equations also model pulse-pulse interactions in wavelength-divisionmultiplexed channels of optical fibre transmission systems. Quasiperiodic wave solutions for the VNLSE in terms of n-dimensional Kleinian functions are presented. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for ncomponent nonlinear Schrödinger equations are found.

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“…Incoherent self-trapping in a biased photorefractive crystal is usually well-described by a set of M coupled nonlinear Schrödinger equations (NLSEs) with saturable nonlinearities [4]. Moreover, in photorefractive media with a drift mechanism of nonlinear response, the Kerr-like nonlinearity is a good model [7]. In the latter case, the equations become integrable, and this allows us to obtain solutions analytically.…”

Section: Introductionmentioning

confidence: 99%

“…It can be shown [7] that, for photorefractive media with a drift mechanism of nonlinear response, a good approximate model describing the propagation of M self-trapped mutually incoherent wave packets in a planar waveguide is the set of NLSEs for a Kerr-type nonlinearity…”

Section: Model Equations and Their Propertiesmentioning

confidence: 99%