Abstract
By applying a simple symmetry reduction on a two-layer liquid model, a nonlocal counterpart of it is obtained. Then, a general form of nonlocal nonlinear Schrödinger (NNLS) equation with shifted parity, charge conjugate and delayed time reversal is obtained by using multi-scale expansion method. Some kinds of elliptic periodic wave solutions of the NNLS equation, which become soliton solutions and kink solutions when the modulus is taken as unity, are obtained by using elliptic function expansion method. Some representative figures of these solutions are given and analyzed in detail. In addition, by carrying out the classical symmetry method on the NNLS equation, not only the Lie symmetry group but also the related symmetry reduction solutions are given.
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Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)
Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139, 7–59 (2017)
Fokas, A.S.: Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 319–324 (2016)
Ma, L.Y., Zhu, Z.N.: N-soliton solution for an integrable nonlocal discrete focusing nonlinear Schrödinger equation. Appl. Math. Lett. 59, 115–121 (2016)
Liu, Y.K., Li, B.: Rogue waves in the (2+1)-dimensional nonlinear Schrödinger equation with a parity-time-symmetric potential. Chin. Phys. Lett. 34, 010202 (2017)
Wen, X.Y., Yan, Z.Y., Yang, Y.Q.: Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Chaos 26(6), 603–151 (2016)
Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 915–946 (2016)
Khare, A., Saxena, A.: Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations. J. Math. Phys. 56, 032104 (2015)
Lou, S.Y., Qiao, Z.J.: Alice–Bob peakon systems. Chin. Phys. Lett. 34(10), 100201 (2017)
Maucher, F., Siminos, E., Krolikowski, W., Skupin, S.: Quasiperiodic oscillations and homoclinic orbits in the nonlinear nonlocal Schrödinger equation. New J. Phys. 15, 083055 (2013)
Cockburn, S.P., Nistazakis, H.E., Horikis, T.P., Kevrekidis, P.G., Proukakis, N.P., Frantzeskakis, D.J.: Matter-wave dark solitons: stochastic versus analytical results. Phys. Rev. Lett. 104, 174101 (2010)
Pertsch, T., Peschel, U., Kobelke, J., Schuster, K., Bartelt, H., Nolte, S., Tunnermann, A., Lederer, F.: Nonlinearity and disorder in fiber arrays. Phys. Rev. Lett. 93, 053901 (2004)
Conti, C., Peccianti, M., Assanto, G.: Observation of optical spatial solitons in a highly nonlocal medium. Phys. Rev. Lett. 92, 113902 (2004)
Lou, S.Y.: Alice–Bob Systems, \(P_s\)-\(T_d\)-\(C\) Principles and Multi-soliton Solutions. arXiv:1603.03975
Lou, S.Y., Huang, F.: Alice–Bob physics: coherent solutions of nonlocal KdV systems. Sci. Rep. 7, 869 (2017)
Jia, M., Lou, S.Y.: Exact \(P_sT_d\) invariant and \(P_sT_d\) symmetric breaking solutions, symmetry reductions and Bäcklund transformations for an AB-KdV system. Phys. Lett. A 382, 1157 (2018)
Li, C.C., Lou, S.Y., Jia, M.: Coherent structure of Alice–Bob modified Korteweg de-Vries equation. Nonlinear Dyn. 93, 1799 (2018)
Tang, X.Y., Liang, Z.F.: A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal. Nonlinear Dyn. 92, 815 (2018)
Tang, X.Y., Liu, S.J., Liang, Z.F., Wang, J.Y.: A general nonlocal variable coefficient KdV equation with shifted parity and delayed time reversal. Nonlinear Dyn. 94(1), 693 (2018)
Tang, X.Y., Liang, Z.F., Hao, X.Z.: Nonlinear waves of a nonlocal modified KdV equation in the atmospheric and oceanic dynamical system. Commun. Nonlinear Sci. 60, 62 (2018)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)
Liu, X.Z., Yu, J., Lou, Z.M.: New interaction solutions from residual symmetry reduction and consistent Riccati expansion of the (\(2+1\))-dimensional Boussinesq equation. Nonlinear Dyn. 92, 1469 (2018)
Liu, X.Z., Yu, J., Lou, Z.M.: New Bäcklund transformations of the (\(2+1\))-dimensional Burgers system related to residual symmetry. Eur. Phys. J. Plus 133, 89 (2018)
Liu, X.Z., Yu, J., Lou, Z.M.: New Bäcklund transformations of the (\(2+1\))-dimensional Bogoyavlenskii equation via localization of residual symmetries. Comput. Math. Appl 76(7), 1669 (2018)
Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1979)
Lou, S.Y.: Symmetry analysis and exact solutions of the \(2+1\) dimensional sine-Gordon system. J. Math. Phys. 41, 6509 (2000)
Acknowledgements
The authors are grateful to the referee, whose comments and suggestions of the earlier version of the paper have led to a substantial clarification and revision of work. This work was supported by the National Natural Science Foundation of China under Grant Nos. 11405110, 11275129 and the Natural Science Foundation of Zhejiang Province of China under Grant No. LY18A050001.
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Liu, Xz., Yu, J. A nonlocal nonlinear Schrödinger equation derived from a two-layer fluid model. Nonlinear Dyn 96, 2103–2114 (2019). https://doi.org/10.1007/s11071-019-04908-8
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DOI: https://doi.org/10.1007/s11071-019-04908-8