New solitons and periodic wave solutions for nonlinear evolution equations

El-Wakil, S.A.; Abdou, M. A.; Elhanbaly, A.

doi:10.1016/j.physleta.2005.12.055

Cited by 83 publications

(49 citation statements)

References 25 publications

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“…Using a developed Jacobi elliptic function expansion method described in [8], we show that in the case of four-order dispersion these traveling wave solutions do not exist. As a byproduct, we regain the results obtained by other authors [8,9] for the case when only the dispersion to the third order is taken into account. …”

supporting

confidence: 84%

“…As it has been emphasized in [8], these solutions are some of the most fundamental objects of study in mathematical physics. Using a developed Jacobi elliptic function expansion method described in [8], we show that in the case of four-order dispersion these traveling wave solutions do not exist. Hence, for ultrashort pulses, it is not necessary to include orders of dispersion which are higher than three to obtain solutions of this type.…”

Section: Introductionmentioning

confidence: 96%

“…By investigating the integrability of a nonlinear PDE, one gains crucial insight into the structure of the equation and nature of its solutions. With the exception of some analytical solutions obtained by well-known methods (inverse scattering method, Hirota's method [5], Jacobi elliptic function expansion method [8]), solitary wave solutions have to be determined numerically [6]. One of the most effective methods is the Fourier Split-Step method which is used in studying the dynamics of short-pulse splitting in dispersive nonlinear media [3,4].…”

Section: Introductionmentioning

confidence: 99%

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Optical solitons in presence of higher-order effects

Van

Dinh

Bang

et al. 2010

Photon. Lett. Poland

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-In this paper, we seek special soliton solutions of so-called Higher-Order Nonlinear Schroedinger Equation, which describes the propagation of light waves in nonlinear optical media. Using a developed Jacobi elliptic function expansion method described in [8], we show that in the case of four-order dispersion these traveling wave solutions do not exist. As a byproduct, we regain the results obtained by other authors [8,9] for the case when only the dispersion to the third order is taken into account.

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supporting

confidence: 84%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Optical solitons in presence of higher-order effects

Van

Dinh

Bang

et al. 2010

Photon. Lett. Poland

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“…the formulas (9), (12) in [12] and (56), (58) in [31]). We demonstrate these for some values of parameters involved in Figures 3-6. …”

Section: Ii52 Gnls With the Four-order Dispersionmentioning

confidence: 99%

Propagation of Ultrashort Pulses in a Nonlinear Medium

Van¹,

Thuan²,

Goldstein³

et al. 2010

CMST

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Abstract:In this paper, using a general propagation equation of ultrashort pulses in an arbitrary dispersive nonlinear medium derived in [8] we study in the specific case of Kerr media. An obtained ultrashort pulse propagation equation which is called Generalized Nonlinear Schrödinger Equation usually has a very complicated form and looking for its solutions is usually a "mission impossible". Theoretical methods to solve this equation are effective only for some special cases. As an example we describe the method of a developed elliptic Jacobi function expansion. Several numerical methods of finding approximate solutions are simultaneously used. We focus mainly on the following methods: Split-Step, Runge-Kutta and Imaginary-time algorithms. Some numerical experiments are implemented for soliton propagation and interacting high order solitons. We consider also an interesting phenomenon: the collapse of solitons.

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“…Recently, Liu [27] obtained some new exact travelling wave solutions of (3) by using the generally projective Riccati equation method. In [28], by using the extended Jacobi elliptic function expansion methods, El-Wakil and Abdou investigated new exact travelling wave solutions of (3) which include a new solitary or shock wave solution and envelope solitary and shock wave solutions. Later on, by using the generalized auxiliary equation method, Abdou [29] studied (3) and obtained some new types of exact travelling wave solutions, including soliton-like solutions, trigonometric function solutions, exponential solutions, and rational solutions.…”

Section: Introductionmentioning

confidence: 99%