Optical Solitons in Presence of Kerr Dispersion and Self-Frequency Shift

Porsezian, K.; Nakkeeran, K.

doi:10.1103/physrevlett.76.3955

Cited by 231 publications

(111 citation statements)

References 19 publications

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“…Thus, solitary wave solutions always exist (in contrast with what is implied in Ref. [7]) provided 3γ 1 + 2γ 2 > 0. For Ω = 0 (i.e., for 3γ 1 + 2γ 2 = 3) the solitary wave solution reduces to…”

mentioning

confidence: 84%

“…(11) into two bilinear equations is impossible in the general case (the direct naïve splitting of Eq. (11) into two equations corresponding to the two expressions in square brackets [13,7] is incorrect, since it gives p 1 = p 2 and q 1 = q 2 for the 2-soliton substitution (12) which is forbidden). Direct analysis of the multilinear equation (11) is not successful either.…”

mentioning

confidence: 99%

“…[7] (which does not fulfill the corresponding nonlinear equation). Note that the existence of N -soliton solutions does not in general imply integrability.…”

mentioning

confidence: 99%

“…[11,13,7]). Substituting this into (17) shows that this is possible only if γ 2 = −3 or ln(G * /G) = const.…”

mentioning

confidence: 99%

“…Ref. [5,7]). This procedure shows that resonances occur at p = −1, 0, 3, 4, and 3 ± 2 −γ 1 / (3γ 1 + 2γ 2 ).…”

mentioning

confidence: 99%

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Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation

1997

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We study solitary wave solutions of the higher order nonlinear Schrödinger equation for the propagation of short light pulses in an optical fiber. Using a scaling transformation we reduce the equation to a two-parameter canonical form. Solitary wave (1-soliton) solutions always exist provided easily met inequality constraints on the parameters in the equation are satisfied. Conditions for the existence of N -soliton solutions (N ≥ 2) are determined; when these conditions are met the equation becomes the modified KdV equation. A proper subset of these conditions meet the Painlevé plausibility conditions for integrability.42.81. Dp, 02.30.Jr, 42.65.Tg, 42.79.Sz The propagation of nonlinear waves in dispersive media is of great interest since nonlinear dispersive systems are ubiquitous in nature. Propagation of ultrashort light pulses in optical fibers is of particular interest because of the common expectation that solitary waves may be of extensive use in telecommunication and even revolutionarize it. The existence of solitary wave solutions implies perfect balance between nonlinearity and dispersion which usually requires rather specific conditions and cannot be established in general. The objective of the present paper is to study the conditions under which the existence of solitary waves is guaranteed for ultrashort pulses.The propagation of light pulses in fibers is well described by the higher order nonlinear Schrödinger equation (HONSE) [1-4], a partial differential equation (PDE) whose right hand side includes the effects of group velocity dispersion, self-phase modulation, third order dispersion, self-steepening, and self-frequency shifting via stimulated Raman scattering, respectively:When the last three terms are omitted this propagation equation for the slowly varying envelope of the electric field, E, reduces to the nonlinear Schrödinger equations (NSE), which is integrable (meaning it not only admits N -solitary wave solutions, but that the evolution of any initial condition is known in principle) [4][5][6]. We call these N -solitary wave solutions N -solitons, and mean by this that the solitary waves scatter elastically and asymptotically preserve their shape upon undergoing collisions, just like true solitons. However, for short duration pulses the last three terms are non-negligible and should be retained. In general, the presence or absence of solitary wave solutions depends on the coefficients α appearing in Eq. (1), and therefore, on the specific nonlinear and dispersive features of the medium. Here, we reduce the HONSE to a two-parameter equation and derive a general solitary wave (1-soliton) solution. We determine conditions when N -soliton solutions exist and display the solutions. We also study the Painlevé plausibility conditions for integrability and show that these are only a proper subset of the conditions necessary for N -soliton solutions to exist. We begin by scaling the HONSE, letting E = b 1 A, z = b 2 ζ, and t = b 3 τ . Substituting into the HONSE we obtainChoosing b 1 = (α 3 1...

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mentioning

confidence: 84%

mentioning

confidence: 99%

“…[7] (which does not fulfill the corresponding nonlinear equation). Note that the existence of N -soliton solutions does not in general imply integrability.…”

mentioning

confidence: 99%

“…[11,13,7]). Substituting this into (17) shows that this is possible only if γ 2 = −3 or ln(G * /G) = const.…”

mentioning

confidence: 99%

“…Ref. [5,7]). This procedure shows that resonances occur at p = −1, 0, 3, 4, and 3 ± 2 −γ 1 / (3γ 1 + 2γ 2 ).…”

mentioning

confidence: 99%

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Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation

1997

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Darboux transformation of a two‐component generalized Sasa–Satsuma equation and explicit solutions

Geng

Wei

et al. 2021

Math Methods in App Sciences

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Under investigation in this paper is a two‐component generalized Sasa–Satsuma equation associated with a 4 × 4 matrix spectral problem. We construct Darboux transformation for the two‐component generalized Sasa–Satsuma equation on basis of the gauge transformation between the Lax pairs. The corresponding N‐fold Darboux transformation is derived in terms of both iterative representation and compact determinant form. As applications of the Darboux transformation together with the limit technique, some interesting solutions including traveling soliton solutions, breather soliton solutions, and rogue wave solutions are explicitly obtained and graphically illustrated.

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Controlling Optical Similaritons in GRIN Waveguide Under the Action of Cubic–Quintic Nonlinear Media

Raju

2024

Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN)

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Optical Solitons in Presence of Kerr Dispersion and Self-Frequency Shift

Cited by 231 publications

References 19 publications

Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation

Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation

Darboux transformation of a two‐component generalized Sasa–Satsuma equation and explicit solutions

Controlling Optical Similaritons in GRIN Waveguide Under the Action of Cubic–Quintic Nonlinear Media

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