Novel higher-order rational solitons and dynamics of the defocusing integrable nonlocal nonlinear Schrödinger equation via the determinants

Abstract We investigate the defocusing integrable nonlocal nonlinear Schrodinger (nNLS) equation using the loop group method and perturbation expansion idea. The associated N -fold Darboux transformation is presented in terms of the simple determinants. As the application of the N -fold Darboux transformation, we find the higher-order rational solitons of the nNLS equation, which exhibit many new types of soliton structures. Moreover, we also numerically study the stability of some rational solitons. The results are useful to explain the corresponding wave phenomena in nonlocal wave models.

[1]  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..

[2]  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.

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

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

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

[6]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

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

[8]  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.

[9]  Mohammad-Ali Miri,et al.  Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[11]  M. Ablowitz,et al.  Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation , 2016 .

[12]  Boris A. Malomed,et al.  Multidimensional solitons: Well-established results and novel findings , 2016, 1603.04097.

[13]  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.

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

[15]  A. Khare,et al.  Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations , 2014, 1405.5267.

[16]  Jürgen Moser,et al.  On a class of polynomials connected with the Korteweg-deVries equation , 1978 .

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

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

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

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

[21]  R. Meinel,et al.  General N-soliton solution of the AKNS class on arbitrary background , 1984 .

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

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

[24]  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.

[25]  Jian Chen,et al.  Generalized Darboux Transformation and Rational Solutions for the Nonlocal Nonlinear Schrödinger Equation with the Self-Induced Parity-Time Symmetric Potential , 2015 .

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

[27]  Zhenya Yan,et al.  Nonlocal general vector nonlinear Schrödinger equations: Integrability, PT symmetribility, and solutions , 2016, Appl. Math. Lett..

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