Multi-dark-dark solitons of the integrable repulsive AB system via the determinants.

We investigate the integrable repulsive AB system and construct its Darboux transformation using the loop group method. The associated N-fold Darboux transformation is found in terms of simple determinants. Moreover, we derive multi-dark-dark solitons of the repulsive AB system with a non-vanishing background through the Darboux transformation with a limit procedure. Particularly, we exhibit the one-, two-, and three-dark-dark solitons. The results will be meaningful for the study of vector multi-dark solitons in many physical systems such as geophysical fluid dynamics and nonlinear optics.

[1]  A. Kamchatnov,et al.  PERIODIC SOLUTIONS AND WHITHAM EQUATIONS FOR THE AB SYSTEM , 1995 .

[2]  A. Degasperis,et al.  Multicomponent integrable wave equations: II. Soliton solutions , 2009, 0907.1822.

[3]  Zhenya Yan,et al.  Vector financial rogue waves , 2011 .

[4]  Zhenya Yan,et al.  Nonautonomous "rogons" in the inhomogeneous nonlinear Schrödinger equation with variable coefficients , 2010, 1009.3731.

[5]  Zhenya Yan,et al.  Modulational instability and higher-order rogue waves with parameters modulation in a coupled integrable AB system via the generalized Darboux transformation. , 2015, Chaos.

[6]  Yong Chen,et al.  Novel higher-order rational solitons and dynamics of the defocusing integrable nonlocal nonlinear Schrödinger equation via the determinants , 2017, Appl. Math. Lett..

[7]  Fabio Baronio,et al.  Rogue-wave bullets in a composite (2+1)D nonlinear medium. , 2016, Optics express.

[8]  John P. Boyd,et al.  Envelope Solitary Waves and Periodic Waves in the AB Equations , 2002 .

[9]  Manuel Mañas,et al.  Darboux transformations for the nonlinear Schrödinger equations , 1996 .

[10]  Zhenya Yan,et al.  Optical rogue waves in the generalized inhomogeneous higher-order nonlinear Schrödinger equation with modulating coefficients , 2013, 1310.3544.

[11]  X. Geng,et al.  The AB Equations and the ∂̄$\bar \partial $-dressing Method in Semi-Characteristic Coordinates , 2013, 1304.4096.

[12]  Zhenya Yan.  Financial Rogue Waves , 2009, 0911.4259.

[13]  G. Swaters,et al.  Finite-amplitude baroclinic instability of a mesoscale gravity current in a channel , 1996 .

[14]  Zhenya Yan,et al.  Interactions of localized wave structures and dynamics in the defocusing coupled nonlinear Schrödinger equations. , 2017, Physical review. E.

[15]  Yong Chen,et al.  Rogue wave solutions of AB system , 2013, Commun. Nonlinear Sci. Numer. Simul..

[16]  Zhenya Yan,et al.  Generalized perturbation (n, M)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Zhenya Yan,et al.  Nonautonomous discrete rogue wave solutions and interactions in an inhomogeneous lattice with varying coefficients , 2012 .

[18]  Zhenya Yan,et al.  Modulational instability, beak-shaped rogue waves, multi-dark-dark solitons and dynamics in pair-transition-coupled nonlinear Schrödinger equations , 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  Zhenya Yan,et al.  Higher-order rational solitons and rogue-like wave solutions of the (2 + 1)-dimensional nonlinear fluid mechanics equations , 2016, Commun. Nonlinear Sci. Numer. Simul..

[20]  Zhenya Yan,et al.  Two-dimensional vector rogue wave excitations and controlling parameters in the two-component Gross–Pitaevskii equations with varying potentials , 2015 .

[21]  Hui-Qin Hao,et al.  Dynamic behaviors of the breather solutions for the AB system in fluid mechanics , 2013 .

[22]  Mark J. McGuinness,et al.  The real and complex Lorenz equations and their relevance to physical systems , 1983 .

[23]  B. Malomed,et al.  Rogue waves, rational solitons, and modulational instability in an integrable fifth-order nonlinear Schrödinger equation. , 2015, Chaos.

[24]  Boling Guo,et al.  Darboux transformation and multi-dark soliton for N-component nonlinear Schrödinger equations , 2013, 1309.1037.

[25]  Zhenya Yan,et al.  Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. , 2016, Chaos.

[26]  Karen Uhlenbeck,et al.  Bäcklund transformations and loop group actions , 1998, math/9805074.

[27]  Zhenya Yan.  Rogon-like solutions excited in the two-dimensional nonlocal nonlinear Schrödinger equation , 2011 .

[28]  Bo Tian,et al.  Integrability aspects and soliton solutions for an inhomogeneous nonlinear system with symbolic computation , 2012 .

[29]  Zhenya Yan,et al.  Three-dimensional rogue waves in nonstationary parabolic potentials. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  J. Pedlosky.  Finite-Amplitude Baroclinic Wave Packets , 1972 .

[31]  P. Kevrekidis,et al.  Solitons in coupled nonlinear Schrödinger models: A survey of recent developments , 2016 .

[32]  Antonio Degasperis,et al.  Multicomponent integrable wave equations: I. Darboux-dressing transformation , 2006, nlin/0610061.

[33]  B. Malomed,et al.  Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: Exact solutions and stability. , 2016, Chaos.