Rogue-wave interaction of the generalized variable-coefficient Hirota–Maxwell–Bloch system in fiber optics

Abstract In this paper, a generalized variable-coefficient Hirota–Maxwell–Bloch system is investigated, which can describe the propagation of optical solitons in an erbium-doped optical fiber. Higher-step generalized Darboux transformation and rogue-wave solutions are obtained. Rogue-wave interaction is analyzed as follows: (1) Variable coefficients in the system affect the shape, background and number of the wave crests and troughs of the first-step rogue waves for the modulus of the normalized slowly varying amplitude of the complex pulse envelope, modulus of the measure of the polarization of the resonant medium and extant population inversion; (2) Variable coefficients in the system affect the shape, background and number of the wave crests and troughs of the second-step rogue-wave interaction. Those phenomena can not be attained through the existing Hirota–Maxwell–Bloch system.

[1]  N. Hoffmann,et al.  Rogue wave observation in a water wave tank. , 2011, Physical review letters.

[2]  Bo Tian,et al.  Symbolic-computation study of the perturbed nonlinear Schrodinger model in inhomogeneous optical fibers [rapid communication] , 2005 .

[3]  Pan Wang,et al.  Integrability and soliton-like solutions for the coupled higher-order nonlinear Schrödinger equations with variable coefficients in inhomogeneous optical fibers , 2014, Commun. Nonlinear Sci. Numer. Simul..

[4]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[5]  K. Nakkeeran,et al.  Optical Soliton Propagation in a Coupled System of the Nonlinear Schrödinger Equation and the Maxwell-Bloch Equations , 1995 .

[6]  V. Matveev,et al.  Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation , 2011 .

[7]  Yufeng Wang,et al.  Soliton dynamics of a discrete integrable Ablowitz-Ladik equation for some electrical and optical systems , 2014, Appl. Math. Lett..

[8]  J. Soto-Crespo,et al.  Rogue waves and rational solutions of the nonlinear Schrödinger equation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  B. Tian,et al.  Double-Wronskian solitons and rogue waves for the inhomogeneous nonlinear Schrödinger equation in an inhomogeneous plasma , 2014 .

[10]  Bo Tian,et al.  Integrability and optical solitons in a generalized variable-coefficient coupled Hirota–Maxwell–Bloch system in fiber optics , 2013 .

[11]  C. Christov,et al.  Physical dynamics of quasi-particles in nonlinear wave equations , 2006, nlin/0612005.

[12]  B. Tian,et al.  Soliton Interactions for the Three‐Coupled Discrete Nonlinear Schrödinger Equations in the Alpha Helical Proteins , 2014 .

[13]  Q. P. Liu,et al.  Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Q. P. Liu,et al.  High‐Order Solutions and Generalized Darboux Transformations of Derivative Nonlinear Schrödinger Equations , 2012, 1205.4369.

[15]  Porsezian,et al.  Optical soliton propagation in an erbium doped nonlinear light guide with higher order dispersion. , 1995, Physical review letters.

[16]  Jingsong He,et al.  Rogue waves of the Hirota and the Maxwell-Bloch equations. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Yasuhiro Ohta,et al.  General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[18]  G. Lamb.  Elements of soliton theory , 1980 .

[19]  Adrian Ankiewicz,et al.  Discrete rogue waves of the Ablowitz-Ladik and Hirota equations. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Efim Pelinovsky,et al.  Physical Mechanisms of the Rogue Wave Phenomenon , 2003 .

[21]  Akira Hasegawa,et al.  Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion , 1973 .

[22]  N. Akhmediev,et al.  Exact first-order solutions of the nonlinear Schrödinger equation , 1987 .

[23]  B. R. Mollow.  Absorption and Emission Line-Shape Functions for Driven Atoms , 1972 .

[24]  B. Jalali,et al.  Optical rogue waves , 2007, Nature.

[25]  S. Mccall,et al.  Self-Induced Transparency by Pulsed Coherent Light , 1967 .

[26]  James P. Gordon,et al.  Experimental observation of picosecond pulse narrowing and solitons in optical fibers (A) , 1980 .

[27]  M. S. Mani Rajan,et al.  Nonlinear tunneling of nonautonomous optical solitons in combined nonlinear Schrödinger and Maxwell–Bloch systems , 2012 .

[28]  Kevin Young,et al.  On Decompositions of the KdV 2-Soliton , 2006, J. Nonlinear Sci..

[29]  Jingsong He,et al.  N-order bright and dark rogue waves in a resonant erbium-doped fiber system , 2012 .

[30]  Adrian Ankiewicz,et al.  Rogue wave triplets , 2011 .

[31]  Adrian Ankiewicz,et al.  Rogue waves and rational solutions of the Hirota equation. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Pierre Gaillard,et al.  On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation , 2010 .

[33]  Bo Tian,et al.  Dynamic behavior of the quantum Zakharov-Kuznetsov equations in dense quantum magnetoplasmas , 2014 .

[34]  Bo Tian,et al.  Dynamics of an integrable Kadomtsev-Petviashvili-based system , 2014, Appl. Math. Lett..