Soliton solutions for the reduced Maxwell–Bloch system in nonlinear optics via the N-fold Darboux transformation

Guo, Rui; Tian, Bo; Wang, Lei

doi:10.1007/s11071-012-0403-5

Cited by 22 publications

(6 citation statements)

References 38 publications

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“…Here, considering the physical significance of the FL equation and the importance of the recent interesting developments in the analysis of PT -symmetric of the NLS and the derivative-type NLS equations, we propose a new reverse space-time nonlocal FL equation as follow i q xt − i q xx + 2 q x − q x q q(−x, −t) + i q = 0, (1.3) which can be derived from a special reduction of the negative flow for the Kaup-Newell (KN) hierarchy. As we all know, the DT method is a powerful and effective mathematical tool to seek new exact soliton solutions [35][36][37][38][39][40]. In Ref.…”

Section: Introductionmentioning

confidence: 99%

Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

2022

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In this paper, we firstly deduce a reverse space-time Fokas-Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation. Next, the determinant representations of one-fold Darboux transformation and N-fold Darboux transformation are expressed in detail by special eigenfunctions of spectral problem. Depending on zero seed solution and nonzero seed solution, exact solutions, including bright soliton solutions, kink solutions, periodic solutions, breather solutions, rogue wave solutions and several types of mixed soliton solutions, can be presented. Furthermore, the dynamical behaviors are discussed through some figures. It should be mentioned that the solutions of nonlocal Fokas-Lenells equation possess new characteristics different from the ones of local case. Besides, we also demonstrate the integrability by providing infinitely many conservation laws. The above results provide an alternative possibility to understand physical phenomena in the field of nonlinear optics, and related fields.

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Section: Introductionmentioning

confidence: 99%

Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

2022

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“…For plane polarized waves, the RMB equations are found integrable and connected with a Zakharov-Shabat scattering problem. Many effective methods, such as the inverse scattering transform (IST) [14,15], the Hirota bilinear method [16][17][18], the Darboux transformation (DT) [19][20][21], the Painlevé analysis [22], etc., have been developed to study the explicit N-soliton solutions of the RMB equations. The RMB equations are one of integrable systems as shown in [23,24] and of course admit other integrable properties including Hamitonian structure and recursion operator [25], the N-degenerate periodic solutions, N-rational solutions and rogue waves [26] as well as interactional solutions [27] by consistent Riccati expansion method.…”

Section: Introductionmentioning

confidence: 99%

Localized excitations and interactional solutions for the reduced Maxwell-Bloch equations

Huang

Chen

2019

Communications in Nonlinear Science and Numerical Simulation

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Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell-Bloch equations. The nonlocal symmetries of the reduced Maxwell-Bloch equations are obtained by the truncated Painlevé expansion approach and the Möbious invariant property. The nonlocal symmetries are localized to a prolonged system by introducing suitable auxiliary dependent variables. The extended system can be closed and a novel Lie point symmetry system is constructed. By solving the initial value problems, a new type of finite symmetry transformations is obtained to derive periodic waves, Ma breathers and breathers travelling on the background of periodic line waves. Then rich exact interactional solutions are derived between solitary waves and other waves including cnoidal waves, rational waves, Painlevé waves, and periodic waves through similarity reductions. In particular, several new types of localized excitations including rogue waves are found, which stem from the arbitrary function generated in the process of similarity reduction. By computer numerical simulation, the dynamics of these localized excitations and interactional solutions are discussed, which exhibit meaningful structures.

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“…The integrability such as the Painlevé test and Lie-algebra-valued differential forms of the RMB equations have been investigated in Refs. [2,3], and the explicit N-soliton solutions of the RMB equations have been respectively studied by the inverse scattering transform, Hirota bilinear technique and Darboux transformation (DT) during the past few decades [4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Periodic and rational solutions of the reduced Maxwell–Bloch equations

Wei

Wang

Geng

2018

Communications in Nonlinear Science and Numerical Simulation

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We investigate the reduced Maxwell-Bloch (RMB) equations which describe the propagation of short optical pulses in dielectric materials with resonant non-degenerate transitions. The general Nth-order periodic solutions are provided by means of the Darboux transformation, and from two different limiting cases of the obtained general periodic solutions, the Nth-order degenerate periodic and Nth-order rational solutions containing several free parameters with compact determinant representations are derived, respectively. Explicit expressions of these solutions from first to second order are presented. Typical nonlinear wave patterns for the four components of the RMB equations such as single-peak, double-peak-double-dip, double-peak and single-dip structures in the second-order rational solutions are shown. This kind of the rational solutions correspond to rogue waves in the reduced Maxwell-Bloch equations.

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Soliton solutions for the reduced Maxwell–Bloch system in nonlinear optics via the N-fold Darboux transformation

Cited by 22 publications

References 38 publications

Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

Localized excitations and interactional solutions for the reduced Maxwell-Bloch equations

Periodic and rational solutions of the reduced Maxwell–Bloch equations

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