Abstract
A (2+1)-dimensional N-coupled nonlinear Schrödinger equation with spatially modulated cubic–quintic nonlinearity and transverse modulation is studied, and vector multipole and vortex soliton solutions are analytically obtained. When the modulation depth q is chosen as 0 and 1, vector multipole and vortex solitons are constructed, respectively. The number of “petals” for the multipole solitons and vortex solitons is related to the value of the topological charge m, and the number of layers in the multipole solitons and vortex solitons is determined by the value of the soliton order number n.
Similar content being viewed by others
References
Zhou, Q., Liu, L., Liu, Y., Yu, H., Yao, P., Wei, C., Zhang, H.: Exact optical solitons in metamaterials with cubic–quintic nonlinearity and third-order dispersion. Nonlinear Dyn. 80, 1365–1371 (2015)
Kong, L.Q., Liu, J., Jin, D.Q., Ding, D.J., Dai, C.Q.: Soliton dynamics in the three-spine \(\alpha \)-helical protein with inhomogeneous effect. Nonlinear Dyn. 87, 83–92 (2017)
Zhou, Q., Zhu, Q., Savescu, M., Bhrawy, A., Biswas, A.: Optical solitons with nonlinear dispersion in parabolic law medium. Proc. Rom. Acad. Ser. A 16, 152–159 (2015)
Zhang, B., Zhang, X.L., Dai, C.Q.: Discussions on localized structures based on equivalent solution with different forms of breaking soliton model. Nonlinear Dyn. 87, 2385–2393 (2017)
Zhou, Q., Liu, L., Liu, Y., Yu, H., Yao, P., Wei, C., Zhang, H.: Exact optical solitons in metamaterials with cubic–quintic nonlinearity and third-order dispersion. Nonlinear Dyn. 80, 1365–1371 (2015)
Wu, H.Y., Jiang, L.H.: Vector Hermite–Gaussian spatial solitons in (2+1)-dimensional strongly nonlocal nonlinear media. Nonlinear Dyn. 83, 713–718 (2016)
Dai, C.Q., Chen, R.P., Wang, Y.Y., Fan, Y.: Dynamics of light bullets in inhomogeneous cubic–quintic–septimal nonlinear media with PT-symmetric potentials. Nonlinear Dyn. 87, 1675–1683 (2017)
Stegeman, G.I., Segev, M.: Optical spatial solitons and their interactions: universality and diversity. Science 286, 1518–1523 (1999)
Dai, C.Q., Wang, X.G.: Light bullet in parity-time symmetric potential. Nonlinear Dyn. 77, 1133–1139 (2014)
Wang, Y.Y., Dai, C.Q., Wang, X.G.: Stable localized spatial solitons in PT-symmetric potentials with power-law nonlinearity. Nonlinear Dyn. 77, 1323–1330 (2014)
Dai, C.Q., Wang, Y.Y.: Spatiotemporal localizations in (3+1)-dimensional PT-symmetric and strongly nonlocal nonlinear media. Nonlinear Dyn. 83, 2453–2459 (2016)
Dai, C.Q., Fan, Y., Zhou, G.Q., Zheng, J., Chen, L.: Vector spatiotemporal localized structures in (3+1)-dimensional strongly nonlocal nonlinear media. Nonlinear Dyn. 86, 999–1005 (2016)
Dai, C.Q., Zhang, J.E.: Controllable dynamical behaviors for spatiotemporal bright solitons on continuous wave background. Nonlinear Dyn. 73, 2049–2057 (2013)
Zhu, H.P., Pan, Z.H.: Vortex soliton in (2+1)-dimensional PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 83, 1325–1330 (2016)
Zhu, H.P., Chen, L., Chen, H.Y.: Hermite–Gaussian vortex solitons of a (3+1)-dimensional partially nonlocal nonlinear Schrodinger equation with variable coefficients. Nonlinear Dyn. 85, 1913–1918 (2016)
Xu, Y.J.: Hollow ring-like soliton and d ipole soliton in (2+1)-dimensional PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 83, 1497–1501 (2016)
Wu, H.Y., Jiang, L.H.: Vector Hermite–Gaussian spatial solitons in (2+1)-dimensional strongly nonlocal nonlinear media. Nonlinear Dyn. 83, 713–718 (2016)
Li, J.T., Zhu, Y., Liu, Q.T., Han, J.Z., Wang, Y.Y., Dai, C.Q.: Vector combined and crossing Kuznetsov–Ma solitons in PT-symmetric coupled waveguides. Nonlinear Dyn. 85, 973–980 (2016)
Dai, C.Q., Wang, Y.Y.: Controllable combined Peregrine soliton and Kuznetsov–Ma soliton in PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 80, 715–721 (2015)
Dai, C.Q., Wang, Y., Liu, J.: Spatiotemporal Hermite–Gaussian solitons of a (3+1)-dimensional partially nonlocal nonlinear Schrodinger equation. Nonlinear Dyn. 84, 1157–1161 (2016)
Pusharov, D.I., Tanev, S.: Bright and dark solitary wave propagation and bistability in the anomalous dispersion region of optical waveguides with third- and fifth-order nonlinearities. Opt. Commun. 124, 354–364 (1996)
Ndzana, F.I., Mohamadou, A., Kofané, T.C.: Modulational instability in the cubic–cquintic nonlinear Schrödinger equation through the variational approach. Opt. Commun. 275, 421–428 (2007)
Dai, C.Q., Wang, Y.: Higher-dimens ional locali zed mode families in parity-time-symmetric potentials with competing nonlinearities. J. Opt. Soc. Am. B 31, 2286–2294 (2014)
Serkin, V.N., Belyaeva, T.L., Alexandrov, I.V., Melchor, G.M.: Novel topological quasi-soliton solutions for the nonlinear cubic–quintic Schrodinger equation model. Proc. SPIE Int. Soc. Opt. Eng. 4271, 292–302 (2001)
Avelar, A.T., Bazeia, D., Cardoso, W.B.: Solitons with cubic and quintic nonlinearities modulated in space and time. Phys. Rev. E 79, 025602 (2009)
Quiroga-Teixeiro, M., Michinel, H.: Stable azimuthal stationary state in quintic nonlinear optical media. J. Opt. Soc. Am. B 14, 2004–2009 (1997)
Agrawal, G.P.: Nonlinear Fiber Optics. Academic, New York (1995)
Gomez-Alcala, R., Dengra, A.: Vector soliton switching by using the cascade connection of saturable absorbers. Opt. Lett. 31, 3137–3139 (2006)
Manakov, S.V.: On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. Phys. JETP 38, 248–253 (1974)
Radhakrishnan, R., Aravinthan, K.: A dark-bright optical soliton solution to the coupled nonlinear Schrödinger equation. J. Phys. A Math. Theor. 40, 13023 (2007)
Zhong, W.P., Belic, M.R., Assanto, G., Malomed, B.A., Huang, T.W.: Self-trapping of scalar and vector dipole solitary waves in Kerr media. Phys. Rev. A 83, 043833 (2011)
Belmonte-Beitia, J., Cuevas, J.: Solitons for the cubic–quintic nonlinear Schrodinger equation with time- and space-modulated coefficients. J. Phys. A Math. Theor. 42, 165201 (2009)
Belmonte-Beitia, J., Perez-Garcia, V.M., Vekslerchik, V., Konotop, V.V.: Localized nonlinear waves in systems with time-and space-modulated nonlinearities. Phys. Rev. Lett. 100, 164102 (2008)
Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1990)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)
Acknowledgements
This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LY17A040011 and LY17F050011) and the National Natural Science Foundation of China (Grant Nos. 11404289 and 11375007).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, YY., Chen, L., Dai, CQ. et al. Exact vector multipole and vortex solitons in the media with spatially modulated cubic–quintic nonlinearity. Nonlinear Dyn 90, 1269–1275 (2017). https://doi.org/10.1007/s11071-017-3725-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-017-3725-5