Pulse Soliton Solutions of the Modified KdV and Born-Infeld Equations

Bogning, Jean Roger

doi:10.4236/ijmnta.2013.22017

Cited by 11 publications

(10 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Using the BDK method to construct exact bright and dark solutions of the generalized higher-order nonlinear Schrödinger equation, we have obtained new exact dark and bright solitary wave solutions. When n 0 =1, the solutions obtained in equations (26) and (32) correspond to those found by Kumar H and Chand F in their studies [30], but for other values of n 0 and for the other solutions construct in this letters, they appear like new analytical solutions of the generalized higher-order nonlinear Schrödinger equation. It is well known that the higher-order nonlinear Schrodinger equation governing the propagation of ultra-short pulse is integrable when the triplet (n 3 , n 4 , n 4 +n 5 )ä{(0, 1, 1), (0, 1, 0), (1, 6, 0), (1, 6, 3)} for n 0 =−n 2 =−2n 1 =1.…”

Section: Resultssupporting

confidence: 76%

Exact bright and dark solitary wave solutions of the generalized higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber

2018

View full text Add to dashboard Cite

Considering linear and nonlinear optical effects like group velocity dispersion, higher-order dispersion, Kerr nonlinearity, self-steepening, stimulated Raman scattering we obtain a higherorder nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber. We construct exact bright and dark solitary wave solutions of the generalized obtained equation, obeying to some constraint relations between coefficient's equation via the Bogning-Djeumen Tchaho-Kofané method (BDKm). The generalized higher-order nonlinear Schrödinger equation is obtained by affecting coefficients n i (i=0, 1, .., 5) to different terms of non modified equation. New solutions are obtained, and the term or higher-order dispersion can be considered as the new selector of solitary wave-type propagating in the higher-order nonlinear optical fiber. IntroductionDuring the last few decades, nonlinear partial differential equations (NPDEs) have seriously draw the attention of the world scientific research. These NPDEs permit to model phenomena in several fields of sciences, notably in physics, such as fluid mechanics, electronics, plasma physics, condensate mater and optical fiber communication [1][2][3][4][5][6][7].The numerical or analytical studies of NPDEs are one of the very important and essential tasks in nonlinear sciences, but the construction of exact solutions plays a very important role, because they can provide much information for a best comprehension of a physical problem. That is why some mathematical methods have been developed to solve NPDEs. We can quote the tanh method, the extended tanh method, the Jacobi's elliptic function algorithm, the Riccati's equation and G′/G method [8][9][10][11][12][13]. The use of these different methods leads to the achievement of travelling wave solutions, periodic solutions and solitary wave solutions. Solitary waves are particular type of solutions of NPDEs with special property. They propagate without changing their shape.The importance of soliton theory in nonlinear science is no more to prove. In area of optical fiber, soliton can be observed, due to the equilibrium between the dispersion and nonlinear effects in the media [14,15]. In communication sciences, solitary wave solutions act as information carrier bits through optical fibers [16,17]. The long distance wired communication across transoceanic and transcontinental distances is now done using optical solitons, and the quantity of information and their transmission speed are unbeatable [18].The propagation of the light in a media is suspected to the dispersion effects and nonlinear effects, which can be of higher-order or not [19,20]. For nonlinear effects, we can note here: Kerr effect or self phase modulation, stimulated brillouin scattering, self-steepening and stimulated Raman scattering [20,21].

show abstract

Section: Resultssupporting

confidence: 76%

Exact bright and dark solitary wave solutions of the generalized higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber

2018

View full text Add to dashboard Cite

show abstract

“…The use of these definitions and the application of Kirchhoff laws to the circuit of nonlinear hybrid electrical line with crosslink capacitor has enabled us to model a set of four nonlinear partial differential equations which describe the dynamics of solitary waves in the line. To construct exact solitary wave solution of each set of four nonlinear partial differential equations, we have used the mathematical methods presented in [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and particularly the Bogning-Djeumen Tchaho-Kofane method [17][18][19][20][21][22]. For one of the set of four nonlinear partial differential equations, we have obtained a solution which is a set of four solitary waves of type (Pulse; Pulse; Pulse; Pulse) and for the other we have obtained a solution which is a set of four solitary waves of type (Kink; Kink; Kink; Kink).…”

Section: Introductionmentioning

confidence: 99%

(Kink; Kink; Kink; Kink) and (Pulse; Pulse; Pulse; Pulse) Solutions of a Set of Four Equations Modeled in a Nonlinear Hybrid Electrical Line with Crosslink Capacitor

Guy

Bogning

2019

PSIJ

Self Cite

View full text Add to dashboard Cite

The physics system that helps us in the study of this paper is a nonlinear hybrid electrical line with crosslink capacitor. Meaning it is composed of two different nonlinear hybrid parts Linked by capacitors with identical constant capacitance. We apply Kirchhoff laws to the circuit of the line to obtain new set of four nonlinear partial differential equations which describe the simultaneous dynamics of four solitary waves. Furthermore, we apply efficient mathematical methods based on the identification of coefficients of basic hyperbolic functions to construct exact solutions of those set of four nonlinear partial differential equations. The obtained results have enabled us to discover that, one of the two nonlinear hybrid electrical line with crosslink capacitor that we have modeled accepts the simultaneous propagation of a set of four solitary waves of type (Pulse; Pulse; Pulse; Pulse), while the other accepts the simultaneous propagation of a set of four solitary waves of type (Kink; Kink; Kink; Kink) when certain conditions we have established are respected. We ameliorate the quality of the signals by changing the sinusoidal waves that are supposed to propagate in the hybrid electrical lines with crosslink capacitor to solitary waves which are propagating in the new nonlinear hybrid electrical lines; we therefore, facilitate the choice of the type of line relative to the type of signal that we want to transmit.

show abstract

“…The mathematical method employed to obtain the results is the Bogning-Djeumen Tchaho-Kofané method (BDKm) [25][26][27][28][29][30][31][32]. The BDKm is a method proposed by J. R. Bogning, C. T. Djeumenand T. C. Kofané in 2012 to construct exact solutions of certain NEEs of the form…”

Section: Introductionmentioning

confidence: 99%

“…(8) intoEq. (7), and taking into account the transformations related to the BDKm[30], gives an equation of the form…”

mentioning

confidence: 99%

Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions

Njikue¹

2018

AJOP

View full text Add to dashboard Cite

In this paper, we modify with an appropriate analytical technique, the characteristics of the optical fiber through the modification of the coefficients of the highly nonlinear partial differential equation, which initially governs the dynamics of the propagation in such a wave guide. The procedure consists to assign arbitrary coefficients to the various terms of the established nonlinear partial differential equation, such as the one that embodies the propagation dynamics in a strongly nonlinear optical fiber and subsequently establishing the constraint equations linking these coefficients and thus the analys is makes it possible to enumerate the criteria for which obtaining the desired solutions is possible. These coefficients are like indicators which characterize the various modifications made in this medium of transmission. The nonlinear evolution equation that served as mathematical model for this study is the higher-order nonlinear Schrödinger equation which better describes the propagation of an ultrafast pulse in an optical fiber. The use of the Bogning-Djeumen Tchaho-Kofané method enabled not only to establish the constraint relations, but also the solitary wave solutions and plane wave solutions. We want through the results obtained in this article to give the specialists of the manufacture of transmission media such as optical fiber, to consider the modification of the properties of this wave guide during manufacture, depending on the type of signal that one wants to propagate in this case notably the solitary wave.

show abstract

Pulse Soliton Solutions of the Modified KdV and Born-Infeld Equations

Abstract: In this work, we use the Bogning-Djeumen Tchaho-Kofané method to look for all solutions of shape Sech n -of the modified KdV and Born-Infeld Equations. n being a real number, we obtain the soliton solutions when n is positive and the non soliton solutions when n is negative.

Cited by 11 publications

References 23 publications

Exact bright and dark solitary wave solutions of the generalized higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber

Exact bright and dark solitary wave solutions of the generalized higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber

(Kink; Kink; Kink; Kink) and (Pulse; Pulse; Pulse; Pulse) Solutions of a Set of Four Equations Modeled in a Nonlinear Hybrid Electrical Line with Crosslink Capacitor

Higher-Order Nonlinear Schrödinger Equation Family in Optical Fiber and Solitary Wave Solutions

Contact Info

Product

Resources

About