Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz-Musslimani equation with PT-symmetric potential.

Starting from a discrete spectral problem, we derive a hierarchy of nonlinear discrete equations which include the Ablowitz-Ladik (AL) equation. We analytically study the discrete rogue-wave (DRW) solutions of AL equation with three free parameters. The trajectories of peaks and depressions of profiles for the first- and second-order DRWs are produced by means of analytical and numerical methods. In particular, we study the solutions with dispersion in parity-time ( PT) symmetric potential for Ablowitz-Musslimani equation. And we consider the non-autonomous DRW solutions, parameters controlling and their interactions with variable coefficients, and predict the long-living rogue wave solutions. Our results might provide useful information for potential applications of synthetic PT symmetric systems in nonlinear optics and condensed matter physics.

[1]  M. Segev,et al.  Theory of Self-Trapped Spatially Incoherent Light Beams , 1997 .

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

[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]  M. Ablowitz,et al.  Nonlinear differential–difference equations and Fourier analysis , 1976 .

[5]  C. Bender,et al.  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , 1997, physics/9712001.

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

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

[8]  Mark J. Ablowitz,et al.  A Nonlinear Difference Scheme and Inverse Scattering , 1976 .

[9]  M. Segev,et al.  Observation of parity–time symmetry in optics , 2010 .

[10]  Mohammad-Ali Miri,et al.  Observation of defect states in PT-symmetric optical lattices. , 2013, Physical review letters.

[11]  Zhenya Yan,et al.  Integrable PT-symmetric local and nonlocal vector nonlinear Schrödinger equations: A unified two-parameter model , 2015, Appl. Math. Lett..

[12]  Xing Zhu,et al.  Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials , 2011, 1110.5108.

[13]  Jianke Yang,et al.  Stability analysis for solitons in PT-symmetric optical lattices , 2012, 1201.2696.

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

[15]  Fajun Yu Nonautonomous discrete bright soliton solutions and interaction management for the Ablowitz-Ladik equation. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Mordechai Segev,et al.  Nonlinearly induced PT transition in photonic systems. , 2013, Physical review letters.

[17]  Z. Musslimani,et al.  Analytical solutions to a class of nonlinear Schrödinger equations with -like potentials , 2008 .

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

[19]  V. Rothos,et al.  Dynamics of the Ablowitz-Ladik soliton train. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  D. Hennig,et al.  Wave transmission in nonlinear lattices , 1999 .

[21]  S. Takeno,et al.  A Propagating Self-Localized Mode in a One-Dimensional Lattice with Quartic Anharmonicity , 1990 .

[22]  M. Ablowitz,et al.  Integrable discrete PT symmetric model. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Daquan Lu,et al.  Solitons supported by complex PT-symmetric Gaussian potentials , 2011 .

[24]  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.

[25]  Zhenya Yan,et al.  Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials , 2015 .

[26]  Fa-Jun Yu,et al.  Nonautonomous rogue waves and 'catch' dynamics for the combined Hirota-LPD equation with variable coefficients , 2016, Communications in nonlinear science & numerical simulation.

[27]  Z. Musslimani,et al.  Optical Solitons in PT Periodic Potentials , 2008 .

[28]  Nick Lazarides,et al.  Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials. , 2012, Physical review letters.

[29]  I. V. Barashenkov,et al.  Breathers in PT-symmetric optical couplers , 2012, 1211.1835.

[30]  M. Ablowitz,et al.  On the solution of a class of nonlinear partial di erence equations , 1977 .

[31]  U. Peschel,et al.  Parity–time synthetic photonic lattices , 2012, Nature.

[32]  Peter S. Lomdahl,et al.  The discrete self-trapping equation , 1985 .

[33]  Qin Zhen-Yun A generalized Ablowitz–Ladik hierarchy, multi-Hamiltonian structure and Darboux transformation , 2008 .

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

[35]  Bikashkali Midya,et al.  Nonlinear localized modes in PT-symmetric Rosen-Morse potential wells , 2013, 1304.2105.

[36]  Boris A. Malomed,et al.  Solitons in nonlinear lattices , 2011 .

[37]  R. Morandotti,et al.  Observation of PT-symmetry breaking in complex optical potentials. , 2009, Physical review letters.

[38]  M. Ablowitz,et al.  Integrable nonlocal nonlinear Schrödinger equation. , 2013, Physical review letters.

[39]  V. Konotop,et al.  Solitons in PT-symmetric nonlinear lattices , 2011, 1104.0276.

[40]  Z. Musslimani,et al.  Beam dynamics in PT symmetric optical lattices. , 2008, Physical review letters.

[41]  Zhenya Yan,et al.  Solitons in a nonlinear Schrödinger equation with PT -symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes , 2015, 1509.05888.

[42]  V. Konotop,et al.  Nonlinear modes in the harmonic PT-symmetric potential , 2012, 1201.6638.