Abstract
The nonlocal defocusing nonlinear Schrödinger (ND-NLS) equation is comparatively studied via the Riemann–Hilbert approach. Firstly, via spectral analysis, the spectral structure of the ND-NLS equation is investigated, which is different to those of the other three NLS-type equations, i.e., the local focusing nonlinear Schrödinger (LF-NLS) equation, the local defocusing nonlinear Schrödinger (LD-NLS) equation and the nonlocal focusing nonlinear Schrödinger (NF-NLS) equation. Secondly, by solving the Riemann–Hilbert problem corresponding to the reflectionless cases, multi-soliton solutions are obtained for the ND-NLS equation. Thirdly, we prove that, if parameters are suitably chosen, the multi-soliton solutions of the ND-NLS equation can be reduced to those of the LF-NLS equation and the LD-NLS equation, respectively. Fourthly, the multi-soliton solutions of the ND-NLS equation are demonstrated to possess repeated singularities generally, but they can also remain analytic for appropriate soliton parameters. Moreover, the multi-soliton dynamics are graphically illustrated using Mathematica symbolic computations. These results show that the solution structure and the nonlinear dynamics in the ND-NLS equation are rather different from those of the LF-NLS equation, the LD-NLS equation and the NF-NLS equation.
Similar content being viewed by others
References
D.J. Benney, A.C. Newell, J. Math. Phys. 46, 133 (1967)
D.J. Benney, G.J. Roskes, Stud. Appl. Math. 48, 377 (1969)
M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981)
V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34, 62 (1972)
M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Stud. Appl. Math. 53, 249 (1974)
M.J. Ablowitz, Z.H. Musslimani, Phys. Rev. Lett. 110, 064105 (2013)
M.J. Ablowitz, Z.H. Musslimani, Nonlinearity 29, 915 (2016)
M.J. Ablowitz, Z.H. Musslimani, Stud. Appl. Math. 139, 7 (2017)
C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)
V.V. Konotop, J.K. Yang, D.A. Zezyulin, Rev. Mod. Phys. 88, 035002 (2016)
T.A. Gadzhimuradov, A.M. Agalarov, Phys. Rev. A 93, 062124 (2016)
M. Li, T. Xu, Phys. Rev. E 91, 033202 (2015)
X. Huang, L.M. Ling, Eur. Phys. J. Plus 131, 148 (2016)
A. Khare, A. Saxena, J. Math. Phys. 56, 032104 (2015)
T. Xu, S. Lan, M. Li, L.L. Li, G.W. Zhang, Phys. D 390, 47 (2019)
J.K. Yang, Phys. Lett. A 383, 328 (2019)
J.K. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, Philadelphia, 2010)
J.K. Yang, D.J. Kaup, J. Math. Phys. 50, 023504 (2009)
D.S. Wang, D.J. Zhang, J.K. Yang, J. Math. Phys. 51, 023510 (2010)
B.L. Guo, L.M. Ling, J. Math. Phys. 53, 073506 (2012)
B.L. Guo, N. Liu, Y.F. Wang, J. Math. Anal. Appl. 459, 145 (2018)
X.G. Geng, J.P. Wu, Wave Motion 60, 62 (2016)
D.S. Wang, X.L. Wang, Nonlinear Anal. RWA 41, 334 (2018)
W.X. Ma, Nonlinear Anal. RWA 47, 1 (2019)
W.X. Ma, J. Geom. Phys. 132, 45 (2018)
W.X. Ma, J. Math. Anal. Appl. 471, 796 (2019)
H.Q. Zhang, Z.J. Pei, W.X. Ma, Chaos Solitons Fract. 123, 429 (2019)
M.J. Ablowitz, A.S. Fokas, Complex Variables: Introduction and Applications (Cambridge University Press, Cambridge, 2003)
A.B.D. Monvel, D. Shepelsky, C. R. Math. 352, 189 (2014)
J.P. Wu, X.G. Geng, Commun. Nonlinear Sci. Numer. Simul. 53, 83 (2017)
J.P. Wu, Nonlinear Dyn. 96, 789 (2019)
V.B. Matveev, M.A. Salle, Darboux Transformation and Solitons (Springer, Berlin, 1991)
Z.X. Zhou, Commun. Nonlinear Sci. Numer. Simulat. 62, 480 (2018)
R. Hirota, The Direct Methods in Soliton Theory (Cambridge University Press, Cambridge, 2004)
A.M. Wazwaz, S.A. El-Tantawy, Nonlinear Dyn. 88, 3017 (2017)
J.J.C. Nimmo, N.C. Freeman, Phys. Lett. A 95, 4 (1983)
W.X. Ma, Y.C. You, Trans. Am. Math. Soc. 357, 1753 (2005)
P.A. Deift, X. Zhou, Ann. Math. 137, 295 (1993)
A.S. Fokas, A Unified Approach to Boundary Value Problems (SIAM, Philadelphia, 2008)
X.G. Geng, H. Liu, J. Nonlinear Sci. 28, 739 (2018)
H. Liu, X.G. Geng, B. Xue, J. Differ. Equ. 265, 5984 (2018)
Acknowledgements
The author is very grateful to the editor and the anonymous referees for their valuable suggestions. The author would also like to thank the support by the Collaborative Innovation Center for Aviation Economy Development of Henan Province.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wu, J. Riemann–Hilbert approach and nonlinear dynamics in the nonlocal defocusing nonlinear Schrödinger equation. Eur. Phys. J. Plus 135, 523 (2020). https://doi.org/10.1140/epjp/s13360-020-00348-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-020-00348-1