Painlevé analysis and bright solitary waves of the higher-order nonlinear Schrödinger equation containing third-order dispersion and self-steepening term

Mihalache, Dumitru; Truta, N; Crasovan, L.-C.

doi:10.1103/physreve.56.1064

Cited by 114 publications

(48 citation statements)

References 35 publications

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“…A thorough discussion on these situations has been given in [6]. In recent years, many authors have analyzed the HNLS equation from different points of view and some interesting results have also been obtained [7][8][9][10][11][12][13]. In this work, we analytically derive both bright and dark solitary wave solutions of HNLS equation under some parametric conditions.…”

Section: Introductionmentioning

confidence: 97%

Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects

Kumar

Chand

2013

Optics & Laser Technology

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Section: Introductionmentioning

confidence: 97%

Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects

Kumar

Chand

2013

Optics & Laser Technology

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“…The integrability of the class of equations (1) has been studied by a number of authors using Painlevé analysis [4,5,6] and other methods [7], with the consistent result that if β 1 , β 2 = 0 there are precisely two integrable cases with bright solitons:…”

Section: B Integrable Casesmentioning

confidence: 99%

Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions

et al. 2003

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Higher order and multicomponent generalizations of the nonlinear Schrödinger equation are important in various applications, e.g., in optics. One of these equations, the integrable Sasa-Satsuma equation, has particularly interesting soliton solutions. Unfortunately the construction of multisoliton solutions to this equation presents difficulties due to its complicated bilinearization. We discuss briefly some previous attempts and then give the correct bilinearization based on the interpretation of the Sasa-Satsuma equation as a reduction of the three-component Kadomtsev-Petviashvili hierarchy. In the process we also get bilinearizations and multi-soliton formulae for a two component generalization of the Sasa-Satsuma equation (the Yajima-Oikawa-Tasgal-Potasek model), and for a (2 + 1)-dimensional generalization.

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“…Of course, one can neglect all the C{ terms of (1) for pulse widths greater than 100 fs, and the resulting equation reduces to a NLS equation with cubic-quintic terms, which has been well studied [9,11], The HONLS models without 7 and c 3 but including the Raman term, i.e. U(\U\ 2 )T, responsible for the self-frequency, a phenomena discovered by Mitschke and Mollenauer [17], have been extensively investigated by various authors [4][5][6][7].…”

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confidence: 99%

Analytical Dark Solitary Wave Solutions for the Higher Order Nonlinear Schrödinger Equation with Cubic-quintic Terms

Hong

2000

Zeitschrift Für Naturforschung A

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By means of the coupled amplitude-phase method we find analytical dark solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic terms describing the effects of quintic nonlinearity on the ultra-short (femtosecond) optical soliton propagation in non-Kerr media. The dark solitary wave solution exists even for the coefficients of quintic terms much larger than those of cubic terms. Optical solitons in Kerr nonlinear media have been the subject of intense current research motivated by their important applications to high-capacity fiber telecommunications and to all optical switches due to their capability of propagating over long distances without attenuation and changing their shapes [1,2]. The dynamics of solitons in Kerr media are in general described by the nonlinear Schrödinger (NLS) family of equations with cubic nonlinear terms [3 -7], However, as one increases the intensity of the incident light to produce shorter (femtosecond) pulses, non-Kerr nonlinearity effects become important and the dynamics of pulses should be described by the NSL family of equations with higher-order nonlinear terms. As such a model, Radhakrshna, Kundu, and Lakshmanan [8] recently proposed the higher order nonlinear Schrödinger (HONLS) equation with cubic-quintic nonlinear terms arising in non-Kerr media. They investigated an integrable system of coupled HONLS equations with some simplifications in the model parameters and found Lax pair, conserved quantities, exact soliton solutions, and analyzed the two-soliton interaction. However, the model [8] along with the general HONLS models proposed in the literature [9 -13], is not completely integrable and can not be solved exactly by the inverse scattering method.To date, exact analytical fundamental solitary wave solutions of the proposed model [8] have not been found. However, analytical and numerical solitary wave solutions have been obtained in the HONLS models in which the effects of quintic nonlinear terms are smaller than the cubic terms [9 -13]. In this work, by using the coupled amplitude-phase formulation [14, 15], we find analytical fundamental dark solitary wave solutions and the constraint equations for the model coefficients. The results show that the dark solitary wave solution exists even though the coefficients of quintic terms are much larger than those of cubic terms.The HONLS equation with cubic-quintic nonlinear non-Kerr terms describing the propagation of femtosecond pulses can be written in the form [8] where r) represents a normalized complex amplitude of the pulse envelope, £ is a normalized distance along the fiber, r is normalized time with the frame of the reference moving along the fiber at the group velocity, and the coefficients c z and 7 are all real. The terms in (1) are: the group velocity dispersion (GVD), self-phase modulation (SPM): The c\ and C2 terms result from including the cubic term in the expansion of the propagation constant and the effects of third order dispersion (TOD) for ultrashort 0932-0784 / 2000 /...

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Painlevé analysis and bright solitary waves of the higher-order nonlinear Schrödinger equation containing third-order dispersion and self-steepening term

Cited by 114 publications

References 35 publications

Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects

Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects

Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions

Analytical Dark Solitary Wave Solutions for the Higher Order Nonlinear Schrödinger Equation with Cubic-quintic Terms

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