Abstract
A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a \(\beta \)-plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.
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References
Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)
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)
Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 915–946 (2016)
Fokas, A.S.: Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 319–324 (2016)
Liu, Y.B., Mihalache, D., He, J.S.: Families of rational solutions of the y-nonlocal Davey–Stewartson II equation. Nonlinear Dyn. 90(4), 2445–2455 (2017)
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)
Ablowitz, M.J., Musslimani, Z.H.: Integrable discrete PT symmetric model. Phys. Rev. E 90, 032912 (2014)
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)
Zhang, Y., Liu, Y.P., Tang, X.Y.: A general integrable three-component coupled nonlocal nonlinear Schrodinger equation. Nonlinear Dyn. 89(4), 2729–2738 (2017)
Lou, S.Y., Huang, F.: Alice–Bob physics: coherent solutions of nonlocal KdV systems. Sci. Rep. 7, 869 (2017)
Tang, X.Y., Zhao, J., Huang, F., Lou, S.Y.: Monopole blocking governed by a modified KdV type equation. Stud. Appl. Math. 122, 295–304 (2009)
Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139, 7–59 (2017)
Jia, M., Gao, Y., Huang, F., Lou, S.Y., Sun, J.L., Tang, X.Y.: Vortices and vortex sources of multiple vortex interaction systems. Nonlinear Anal. Real Word Appl. 13, 2079 (2012)
Dudley, J.M., Genty, G., Dias, F., Kibler, B., Akhmediev, N.: Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation. Opt. Express 23, 21497 (2009)
Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)
Khare, A., Saxena, A.: Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations. J. Math. Phys. 56, 032104 (2015)
Luo, D.H., Li, J.P.: Interaction between a slowly moving planetary-scale dipole envelop Rossby soliton and a wavenumber-two topography in a forced higher order nonlinear Schrödinger equation. Adv. Atmos. Sci. 19, 239–256 (2001)
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The authors acknowledge the financial support by the National Natural Science Foundation of China (Nos. 11675055 and 11475052) and Shanghai Knowledge Service Platform for Trustworthy Internet of Things (No. ZF1213).
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Tang, XY., Liang, ZF. A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal. Nonlinear Dyn 92, 815–825 (2018). https://doi.org/10.1007/s11071-018-4092-6
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DOI: https://doi.org/10.1007/s11071-018-4092-6