Rogue waves for a discrete (2+1)-dimensional Ablowitz-Ladik equation in the nonlinear optics and Bose-Einstein condensation

Abstract Under investigation in this paper is a discrete (2+1)-dimensional Ablowitz-Ladik equation, which is used to model the nonlinear waves in the nonlinear optics and Bose-Einstein condensation. Employing the Kadomtsev-Petviashvili hierarchy reduction, we obtain the rogue wave solutions in terms of the Gramian. We graphically study the first-, second- and third-order rogue waves with the influence of the focusing coefficient and coupling strength. When the value of the focusing coefficient increases, both the peak of the rogue wave and background decrease. When the value of the coupling strength increases, the rogue wave raises and decays in a shorter time. High-order rogue waves are exhibited as one single highest peak and some lower humps, and such lower humps are shown as the triangular and circular patterns.

[1]  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 .

[2]  Stephane Lafortune,et al.  The higher-dimensional Ablowitz–Ladik model: From (non-)integrability and solitary waves to surprising collapse properties and more exotic solutions , 2009, 0907.1386.

[3]  J. Bilbault,et al.  Observation of nonlinear localized modes in an electrical lattice. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  Yi-Tian Gao,et al.  Bilinear forms and solitons for a generalized sixth-order nonlinear Schrödinger equation in an optical fiber , 2017 .

[5]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[6]  Lei Hu,et al.  Breather-to-soliton transition for a sixth-order nonlinear Schrödinger equation in an optical fiber , 2018, Appl. Math. Lett..

[7]  広田 良吾,et al.  The direct method in soliton theory , 2004 .

[8]  Yasuhiro Ohta,et al.  General rogue waves in the focusing and defocusing Ablowitz–Ladik equations , 2014 .

[9]  Fajun Yu,et al.  Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz-Musslimani equation with PT-symmetric potential. , 2017, Chaos.

[10]  N. Akhmediev,et al.  Waves that appear from nowhere and disappear without a trace , 2009 .

[11]  Jakub Zakrzewski,et al.  Non-standard Hubbard models in optical lattices: a review , 2014, Reports on progress in physics. Physical Society.

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

[13]  Xi-Yang Xie,et al.  Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers. , 2017, Chaos.

[14]  V. Konotop,et al.  Matter rogue waves , 2009 .

[15]  Hui-Qin Hao,et al.  Breathers and localized solitons for the Hirota–Maxwell–Bloch system on constant backgrounds in erbium doped fibers , 2014 .

[16]  Hui-Qin Hao,et al.  N-fold Darboux transformation and discrete soliton solutions for the discrete Hirota equation , 2018, Appl. Math. Lett..

[17]  Oleksiy O. Vakhnenko,et al.  Physically corrected Ablowitz-Ladik model and its application to the Peierls-Nabarro problem , 1995 .

[18]  Kwok Wing Chow,et al.  Rogue wave modes for a derivative nonlinear Schrödinger model. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  J. Dudley,et al.  Rogue-wave-like characteristics in femtosecond supercontinuum generation. , 2009, Optics letters.

[20]  Yi-Tian Gao,et al.  Solitons for the (3+1)-dimensional variable-coefficient coupled nonlinear Schrödinger equations in an optical fiber , 2017 .

[21]  Yi-Tian Gao,et al.  Integrability, solitons, periodic and travelling waves of a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles , 2017 .

[22]  Stefan Nolte,et al.  Realization of reflectionless potentials in photonic lattices. , 2011, Physical review letters.

[23]  N. Akhmediev,et al.  Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy. , 2017, Physical review. E.

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

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

[26]  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.

[27]  Zhang Jin-Liang,et al.  Exact solutions and linear stability analysis for two-dimensional Ablowitz–Ladik equation , 2014 .

[28]  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..

[29]  Zhong-Zhou Lan,et al.  Solitons, breather and bound waves for a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide , 2017 .

[30]  Mark J. Ablowitz,et al.  Nonlinear differential−difference equations , 1975 .