Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation

Abstract A nonlocal derivative nonlinear Schrodinger equation is discussed. By constructing its Darboux transformations of degree 2n, the explicit expressions of new solutions are derived from zero seed solutions. Usually the derived solutions of this nonlocal equation may have singularities. However, it is shown that the solutions of the nonlocal derivative nonlinear Schrodinger equation can be globally defined and bounded for all (x, t) if the eigenvalues and the parameters characterizing the ratio of the two entries of the solutions of the Lax pair are chosen properly.

[1]  V. Zakharov,et al.  On the integrability of classical spinor models in two-dimensional space-time , 1980 .

[2]  Kenji Imai,et al.  Generalization of the Kaup-Newell Inverse Scattering Formulation and Darboux Transformation , 1999 .

[3]  H. Steudel,et al.  The hierarchy of multi-soliton solutions of the derivative nonlinear Schrödinger equation , 2003 .

[4]  E. Mjølhus,et al.  On the modulational instability of hydromagnetic waves parallel to the magnetic field , 1976, Journal of Plasma Physics.

[5]  A. S. Fokas,et al.  Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation , 2016 .

[6]  David H. Sattinger,et al.  Gauge theory of Ba¨cklund transformations, II , 1987 .

[7]  Jianke Yang,et al.  Nonlinear waves in PT -symmetric systems , 2016, 1603.06826.

[8]  M. Ablowitz,et al.  Integrable Nonlocal Nonlinear Equations , 2016, 1610.02594.

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

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

[11]  Liming Ling,et al.  Soliton solutions for the nonlocal nonlinear Schrödinger equation , 2016 .

[12]  David J. Kaup,et al.  An exact solution for a derivative nonlinear Schrödinger equation , 1978 .

[13]  T. Gadzhimuradov,et al.  Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear Schrödinger equation , 2016 .

[14]  Li-Yuan Ma,et al.  N-soliton solution for an integrable nonlocal discrete focusing nonlinear Schrödinger equation , 2016, Appl. Math. Lett..

[15]  Zixiang Zhou,et al.  Darboux Transformations in Integrable Systems , 2005 .

[16]  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.

[17]  Zuo-Nong Zhu,et al.  Solitons and dynamics for a general integrable nonlocal coupled nonlinear Schrödinger equation , 2017, Commun. Nonlinear Sci. Numer. Simul..

[18]  S. Lou,et al.  Alice-Bob Physics: Coherent Solutions of Nonlocal KdV Systems , 2016, Scientific Reports.

[19]  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.

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