Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms

Hong, Woo-Pyo

doi:10.1016/s0030-4018(01)01267-6

Cited by 105 publications

(23 citation statements)

References 24 publications

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“…This means that it is possible to find both solitary wave and multi-soliton solutions [3]. However, as one increases the intensity of the incident light power to produce shorter (femtosecond) pulses, non-Kerr nonlinearity effects become important, and the dynamics of pulses should be described by the nonlinear Schrödinger family of equations with higher order nonlinear terms [4]. Hence, it is very important that all higher order effects be considered in the propagation of femtosecond pulses [5].…”

Section: Introductionmentioning

confidence: 99%

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

Zheng

Shang

2014

Math Methods in App Sciences

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Communicated by S. A. MessaoudiThis paper is concerned with the generalized nonlinear Schrödinger equation with parabolic law and dual-power law. Abundant explicit and exact solutions of the generalized nonlinear Schrödinger equation with parabolic law and dualpower law are derived uniformly by using the first integral method. These exact solutions are include that of extended hyperbolic function solutions, periodic wave solutions of triangle functions type, exponential form solution, and complex hyperbolic trigonometric function solutions and so on. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial DEs.Recently, many authors have studied the dynamics of bright solitons and dark solitons that are governed by the generalized nonlinear Schrödinger equations, and some interesting results have been obtained [6,[10][11][12][13][14][15][16].In [6,10,11], some authors have studied the dynamics of bright optical solitons that are governed by the NLSE with constant and variable coefficients. In [12], the author studied dark solitons, which are known as topological optical solitons in the context of nonlinear optics. Compared with the bright soliton, which is a pulse on a zero-intensity background, the dark soliton appears as an intensity dip in an infinitely extended constant background [15]. In [16], dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law were obtained by using ansatz method.The aim of this paper is to supply a unified method for constructing a series of explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual-power law. The first integral method is employed to investigate the generalized nonlinear Schrödinger equation with parabolic law and dual-power law. Through an exhaustive analysis and discussion for different parameters, we uniformly construct a series of explicit exact solutions to Eqs. (1) and (2). Compared with ansatz method used in [10,16], the first integral method gives abundant explicit exact solitary wave solutions, periodic wave solutions of triangle function, extended hyperbolic function solutions, complex hyperbolic trigonometric function solutions, and so on. Especially, explicit exact solutions obtained in the present paper include the results obtained in [10,16] as a special case. Many known results in other previous literature are also improved and completed greatly.The rest of this paper is organized as follows. In Section 2, the outline of the first integral method will be given. In Section 3 , the method is employed to seek the explicit and exact solutions of the generalized nonlinear Schrödinger equation with parabolic law. In Section 4, the method is employed to seek the explicit and exact solutions of the generalized nonlinear Schrödinger equation with dual-power law. In the last section, some conclusion is given.We determine polynomials˛.X/,ˇ.X/, a i .X/.i D 0, 1, 2, / from Eq. (...

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Section: Introductionmentioning

confidence: 99%

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

Zheng

Shang

2014

Math Methods in App Sciences

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“…This means that it is possible to find both solitary wave and multi-soliton solutions [8]. However, as one increases the intensity of the incident light power to produce shorter (femtosecond) pulses, non-Kerr nonlinearity effects become important and the dynamics of pulses should be described by the NLS family of equations with higher order nonlinear terms [9]. Hence, it is very important that all higher order effects be considered in the propagation of femtosecond pulses [10].…”

Section: Introductionmentioning

confidence: 99%

Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities

Triki

Biswas

2011

Math. Meth. Appl. Sci.

107

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Communicated by S. GeorgievThe dynamics of dark solitons is studied within the framework of a generalized nonlinear Schrödinger equation. The specific cases of parabolic law and dual-power law nonlinearity are considered. The solitary wave ansatz method is used to carry out the integration. All the physical parameters in the soliton solutions are obtained as functions of the dependent model coefficients. Parametric conditions for the existence of envelope solitons are given.

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“…The dynamics of such systems should be described by the NLSE with higher-order terms such as third-order dispersion, self-steepening, and selffrequency shift [5,6]. Moreover, in some physical situations cubic-quintic nonlinear terms arise [7,8], due to non-Kerr nonlinearities, from a nonlinear correction to the refractive index of a medium. In general, unlike the NLSE, these models with non-Kerr effects are not completely integrable and cannot be solved exactly by the inverse scattering transform method [9].…”

Section: Introductionmentioning

confidence: 99%

Chirped femtosecond solitons and double-kink solitons in the cubic-quintic nonlinear Schrödinger equation with self-steepening and self-frequency shift

et al. 2011

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We demonstrate that the competing cubic-quintic nonlinearity induces propagating solitonlike dark(bright) solitons and double-kink solitons in the nonlinear Schrödinger equation with self-steepening and self-frequency shift. Parameter domains are delineated in which these optical solitons exist. Also, fractional-transform solitons are explored for this model. It is shown that the nonlinear chirp associated with each of these optical pulses is directly proportional to the intensity of the wave and saturates at some finite value as the retarded time approaches its asymptotic value. We further show that the amplitude of the chirping can be controlled by varying the self-steepening term and self-frequency shift.

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Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms

Cited by 105 publications

References 24 publications

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities

Chirped femtosecond solitons and double-kink solitons in the cubic-quintic nonlinear Schrödinger equation with self-steepening and self-frequency shift

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