Soliton and breather interactions for a coupled system

Abstract.Under investigation in this paper is a higher-order nonlinear Schrödinger-Maxwell-Bloch system with the sextic term which might describe the ultrashort optical pulses, up to the attosecond duration, in an erbium-doped fiber. We derive the Lax pair and Darboux transformation, which are both related to $ \beta$β and $ \omega$ω , the coefficient for the higher-order terms and the detuning of the atomic transition frequency from the incoming radiation frequency, respectively. Bright and dark solitons and breathers are constructed by virtue of the Darboux transformation. Besides, based on the breather solutions, we get the first-order rogue wave solutions via the limiting procedure. We see that both $ \beta$β and $ \omega$ω affect velocities and widths of the solitons, also decide whether the solution is a bright or dark soliton while have no effect on the periods of the two types of breathers. The interaction between the solitons is elastic, and the periodic structure comes into being with the interaction between the two solitons when the velocities of the two interacted solitons are equal to each other. With different eigenvalues, different interaction forms between the two types of the breathers are obtained. Interaction between the two breathers with the adjacent frequencies forms the bound breather with a periodic attractive-repulsive structure. Especially, two near-degenerate cases of the interactions are also exhibited.

[1]  Xin-Yi Gao,et al.  Looking at a nonlinear inhomogeneous optical fiber through the generalized higher-order variable-coefficient Hirota equation , 2017, Appl. Math. Lett..

[2]  C. Finot,et al.  Higher-order modulation instability in nonlinear fiber optics. , 2011, Physical review letters.

[3]  Bo Tian,et al.  Solitons for the (2+1)-dimensional Konopelchenko–Dubrovsky equations , 2018 .

[4]  Ying Liu,et al.  Bound vector solitons and soliton complexes for the coupled nonlinear Schrödinger equations. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  B. Tian,et al.  Vector bright soliton interactions of the coupled Sasa–Satsuma equations in the birefringent or two-mode fiber , 2018, Wave Motion.

[6]  Bo Tian,et al.  Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schrödinger equations in an alpha helical protein , 2019, Commun. Nonlinear Sci. Numer. Simul..

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

[8]  K. Porsezian,et al.  SOLITONS IN AN ERBIUM-DOPED NONLINEAR FIBRE MEDIUM WITH STIMULATED INELASTIC SCATTERING , 1995 .

[9]  Xin-Yi Gao.  Bäcklund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid mechanics , 2015 .

[10]  Yi-Tian Gao,et al.  Rogue-wave interaction for a higher-order nonlinear Schrödinger–Maxwell–Bloch system in the optical-fiber communication , 2014 .

[11]  Yair Zarmi,et al.  Wave interactions and the analysis of the perturbed Burgers equation , 2005 .

[12]  Yamada,et al.  Coexistence of a self-induced-transparency soliton and a nonlinear Schrödinger soliton in an erbium-doped fiber. , 1991, Physical review. A, Atomic, molecular, and optical physics.

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

[14]  Bo Tian,et al.  Rogue waves and solitons of the coherently-coupled nonlinear Schrödinger equations with the positive coherent coupling , 2018, Physica Scripta.

[15]  K. Porsezian.  Optical solitons in some SIT type equations , 2000 .

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

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

[18]  I. Christov.  Enhanced generation of attosecond pulses in dispersion-controlled hollow-core fiber , 1999 .

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

[20]  Bo Tian,et al.  Mixed lump-kink and rogue wave-kink solutions for a (3 + 1) -dimensional B-type Kadomtsev-Petviashvili equation in fluid mechanics , 2018 .

[21]  N. Akhmediev,et al.  Modulation instability and periodic solutions of the nonlinear Schrödinger equation , 1986 .

[22]  Bo Tian,et al.  Dark-bright solitons and semirational rogue waves for the coupled Sasa-Satsuma equations. , 2018, Physical review. E.

[23]  Multi-soliton solutions of a coupled system of the nonlinear Schrödinger equation and the Maxwell-Bloch equations , 1994 .

[24]  A Ankiewicz,et al.  Infinite hierarchy of nonlinear Schrödinger equations and their solutions. , 2016, Physical review. E.

[25]  B. Tian,et al.  Rogue-wave solutions for a discrete Ablowitz–Ladik equation with variable coefficients for an electrical lattice , 2018, Nonlinear Dynamics.

[26]  Govind P. Agrawal,et al.  Nonlinear Fiber Optics , 1989 .

[27]  Lei Liu,et al.  Rogue waves for a variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics , 2018, Comput. Math. Appl..

[28]  K. Porsezian,et al.  Solitons in random nonuniform erbium doped nonlinear fiber media , 1995 .

[29]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[30]  Bo Tian,et al.  Lie group analysis, analytic solutions and conservation laws of the (3 + 1)-dimensional Zakharov-Kuznetsov-Burgers equation in a collisionless magnetized electron-positron-ion plasma , 2018, The European Physical Journal Plus.

[31]  B. Tian,et al.  Rogue waves for a generalized nonlinear Schrödinger equation with distributed coefficients in a monomode optical fiber , 2018 .

[32]  Yan‐Chow Ma,et al.  The Perturbed Plane‐Wave Solutions of the Cubic Schrödinger Equation , 1979 .

[33]  Bo Tian,et al.  Multi-soliton interaction of a generalized Schrödinger-Boussinesq system in a magnetized plasma , 2017 .

[34]  Kimura,et al.  Self-induced-transparency solitons in an erbium-doped fiber waveguide. , 1992, Physical review. A, Atomic, molecular, and optical physics.

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

[36]  Adrian Ankiewicz,et al.  Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  D. Solli,et al.  Recent progress in investigating optical rogue waves , 2013 .

[38]  W. Moslem,et al.  Langmuir rogue waves in electron-positron plasmas , 2011 .

[39]  A Ankiewicz,et al.  Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Yamada,et al.  Coexistence of self-induced transparency soliton and nonlinear Schrödinger soliton. , 1991, Physical review letters.