Stationary waves in a third-order nonlinear Schrödinger equation

Gromov, E. M.; Tyutin, V. V.

doi:10.1016/s0165-2125(97)00060-7

Cited by 13 publications

(8 citation statements)

References 11 publications

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“…23 for ␥ϭ0 and ␤ and in Ref. 24 for ␥(q␤ Ϫ3␥␣) 0. Soliton solutions with time-independent shift of frequency were found in the NSE-3 for the following three cases: ͑1͒ at the zero dispersion point ͑ZDP͒ corresponding to the zero second-order linear dispersion (qϭ0), Ref.…”

Section: Short Optical Solitonsmentioning

confidence: 92%

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Short optical solitons in fibers

Gromov

Talanov

2000

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The dynamics of short (of the order of a few wave periods) intense optical pulses and interaction of short optical solitons in fibers are considered within the framework of the third-order nonlinear Schrodinger equation. It is shown that an initial pulse tends to one or a few short solitons plus a linear quasiperiodic wave when the third-order linear dispersion and nonlinear dispersion have parameters of the same sign. The number and parameters of the solitons depend on the magnitudes of initial pulse parameters. Interaction of short optical solitons having different amplitudes is accompanied by radiation of part of the wave field from the area of interaction, by an increase of the soliton with larger amplitude, and a decrease of the soliton with a smaller one. (c) 2000 American Institute of Physics.

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Section: Short Optical Solitonsmentioning

confidence: 92%

“…19 Still another analytical method of analysis of the NSE-3 based on reducing it to ordinary differential equations was employed in Refs. [20][21][22][23][24]. Stationary waves with time-dependent shift of frequency were found by Anderson and Lisak 20 for ␥ϭ0 and ␤ϭ, in Ref.…”

Section: Short Optical Solitonsmentioning

confidence: 99%

Short optical solitons in fibers

Gromov

Talanov

2000

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…18 It was shown with the soliton perturbation theory 32 that the one-parameter family of embedded solitons with 0 Ͻ ͉␣͉ + ͉␥͉ Ӷ ␤ is linearly and nonlinearly stable in the time evolution of arbitrary localized initial data. This one-parameter family of embedded solitons ͑2.2͒-͑2.4͒ gives an exact solution of the third-order NLS Eq.…”

Section: ͑23͒mentioning

confidence: 99%

Stability analysis of embedded solitons in the generalized third-order nonlinear Schrödinger equation

Pelinovsky

Yang

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study the generalized third-order nonlinear Schrodinger (NLS) equation which admits a one-parameter family of single-hump embedded solitons. Analyzing the spectrum of the linearization operator near the embedded soliton, we show that there exists a resonance pole in the left half-plane of the spectral parameter, which explains linear stability, rather than nonlinear semistability, of embedded solitons. Using exponentially weighted spaces, we approximate the resonance pole both analytically and numerically. We confirm in a near-integrable asymptotic limit that the resonance pole gives precisely the linear decay rate of parameters of the embedded soliton. Using conserved quantities, we qualitatively characterize the stable dynamics of embedded solitons.

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“…However, solitary waves of Eq. (1) which are embedded inside the linear spectrum do exist in certain parameter regimes [6][7][8][9], and such waves are called embedded solitons [10,11]. To see why a solitary wave of Eq.…”

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confidence: 99%

Stable Embedded Solitons

2003

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Stable embedded solitons are discovered in the generalized third-order nonlinear Schrödinger equation. When this equation can be reduced to a perturbed complex modified KdV equation, we developed a soliton perturbation theory which shows that a continuous family of sech-shaped embedded solitons exist and are nonlinearly stable. These analytical results are confirmed by our numerical simulations. These results establish that, contrary to previous beliefs, embedded solitons can be robust despite being in resonance with the linear spectrum. [PACS: 42.65.Tg, 05.45.Yv.] Pulse propagation in optical fibers attracts a lot of attention these days. Pico-second pulses are well described by the nonlinear Schrödinger (NLS) equation which accounts for the second-order dispersion and self-phase modulation. But for femtosecond pulses, other physical effects such as the third-order dispersion and selfsteepening become non-negligible. The physical model which incorporates these additional effects is [1,2] (1)Here ψ is the envelope of the electric field, z is the distance, τ is the retarded time, β and µ are the selfsteepening coefficients [1]. All quantities have been normalized. For optical pulses, β = µ [1,2]. In this paper, we allow β and µ to be different for the sake of mathematical analysis. The Raman effect, which is dissipative in nature, is also non-negligible [1,2]. It is not included in the model (1) because we want to focus our attention on the other physical effects in this paper. The third-order dispersion term in Eq.(1) is significant because it qualitatively changes the linear dispersion relation of Eq. (1). Its effect on the NLS soliton is to generate continuous-wave radiation and causes the soliton to decay [1][2][3][4][5]. However, solitary waves of Eq. (1) which are embedded inside the linear spectrum do exist in certain parameter regimes [6][7][8][9], and such waves are called embedded solitons [10,11]. To see why a solitary wave of Eq. (1) has to be an embedded soliton, we substitute solitary waves of the form Ψ(τ − V z)e iλz into (1), where velocity V and frequency λ are constants. We readily find that for any frequency λ, the linear equation for Ψ allows oscillatory solutions. Thus, all λ lies in the continuous spectrum of Eq. (1), hence the solitary wave must be an embedded soliton. Stability of these embedded solitons is clearly an important issue. Previous analytical studies have shown that if embedded solitons are isolated in a conservative system, they are at most semi-stable, i.e., the perturbed soliton persists or decays depending on whether the initial energy is higher or lower than that of the embedded soliton [10,12,13]. A physical explanation for it is as follows [10]. If the initial state has energy higher than the embedded soliton, it just sheds extra energy through tail radiation and asymptotically approaches this embedded soliton; but if the initial state has lower energy than the embedded soliton, the energy loss (through radiation) drives the solution away from the embedded soliton, re...

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Stationary waves in a third-order nonlinear Schrödinger equation

Cited by 13 publications

References 11 publications

Short optical solitons in fibers

Short optical solitons in fibers

Stability analysis of embedded solitons in the generalized third-order nonlinear Schrödinger equation

Stable Embedded Solitons

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