Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential.

The integrable nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential [M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 110, 064105 (2013)] is investigated, which is an integrable extension of the standard nonlinear Schrödinger equation. Its novel higher-order rational solitons are found using the nonlocal version of the generalized perturbation (1,N-1)-fold Darboux transformation. These rational solitons illustrate abundant wave structures for the distinct choices of parameters (e.g., the strong and weak interactions of bright and dark rational solitons). Moreover, we also explore the dynamical behaviors of these higher-order rational solitons with some small noises on the basis of numerical simulations.

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

[2]  Frédéric Dias,et al.  The Peregrine soliton in nonlinear fibre optics , 2010 .

[3]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

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

[5]  C. Gu.  Bäcklund Transformations and Darboux Transformations , 1995 .

[6]  Yuri S. Kivshar,et al.  Optical Solitons: From Fibers to Photonic Crystals , 2003 .

[7]  Norman J. Zabusky,et al.  Soliton , 2010, Scholarpedia.

[8]  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.

[9]  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.

[10]  Higinio Mora-Mora,et al.  μ-MAR: Multiplane 3D Marker based Registration for depth-sensing cameras , 2015, Expert Syst. Appl..

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

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

[13]  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.

[14]  Triangular rogue wave cascades. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[16]  Yasuhiro Ohta,et al.  Rogue waves in the Davey-Stewartson I equation. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  B. Malomed,et al.  Spatiotemporal optical solitons , 2005 .

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

[19]  W. Perrie,et al.  Nonlinear Ocean Waves , 1997 .

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

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

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

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

[24]  Lihong Wang,et al.  Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  B. M. Fulk.  MATH , 1992 .

[26]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

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

[29]  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.

[30]  S. Manakov,et al.  On the complete integrability of a nonlinear Schrödinger equation , 1974 .

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

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

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

[34]  Adrian Ankiewicz,et al.  Circular rogue wave clusters. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[36]  Y. Nakamura,et al.  Observation of Peregrine solitons in a multicomponent plasma with negative ions. , 2011, Physical review letters.

[37]  Harald E. Krogstad,et al.  Oceanic Rogue Waves , 2008 .

[38]  D. H. Peregrine,et al.  Water waves, nonlinear Schrödinger equations and their solutions , 1983, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

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

[40]  N. Akhmediev,et al.  Are rogue waves robust against perturbations , 2009 .

[41]  Yasuhiro Ohta,et al.  Dynamics of rogue waves in the Davey–Stewartson II equation , 2012, 1212.0152.

[42]  Karsten Trulsen,et al.  NOTE ON BREATHER TYPE SOLUTIONS OF THE NLS AS MODELS FOR FREAK-WAVES , 1999 .

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

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

[45]  Min Li,et al.  Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  V. Matveev,et al.  Multi-rogue waves solutions: from the NLS to the KP-I equation , 2013 .

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

[48]  V. Zakharov,et al.  Freak waves as nonlinear stage of Stokes wave modulation instability , 2006 .

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

[50]  B. Jalali,et al.  Active control of rogue waves for stimulated supercontinuum generation. , 2008, Physical review letters.

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

[52]  Andrew G. Glen,et al.  APPL , 2001 .