Solitary wave solutions for nonlinear fractional Schrödinger equation in Gaussian nonlocal media.(English)Zbl 1410.35222

Summary: This article is devoted to the study of nonlinear fractional Schrödinger equation with a Gaussian nonlocal response. We firstly prove the existence of solitary wave solutions by using the variational method and Mountain Pass Theorem. Numerical simulations are presented to verify the findings of the existence theorem. And we also investigate the impacts of Gaussian nonlocal response and fractional-order derivatives on the solitary waves, which enable us to perform control experiments for the development of rogue waves in quantum mechanics and optics.

MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35R11 Fractional partial differential equations 35C08 Soliton solutions 35A15 Variational methods applied to PDEs 81V80 Quantum optics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
Full Text:

References:

 [1] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, (2016), Elsevier [2] Miller, K. S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002 [3] Felmer, P.; Quaas, A.; Tan, J., Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142, 1237-1262, (2012) · Zbl 1290.35308 [4] Secchi, S., Ground state solutions for nonlinear fractional Schrödinger equations in $$\mathbb{R}^N$$, J. Math. Phys., 54, (2013) · Zbl 1281.81034 [5] Yang, X. J.; Gao, F.; Srivastavacd, H. M., A new computational approach for solving nonlinear local fractional pdes, J. Comput. Appl. Math., 339, 285-296, (2018) · Zbl 06867159 [6] Yang, X. J.; Srivastava, H. M.; Machado, J. A., A new fractional derivative without singular kernel: application to the modelling of the steady heat flow, Therm. Sci., 20, 2, 753-756, (2016) [7] Yang, X. J.; Machado, J. A.T., A new fractional operator of variable order: application in the description of anomalous diffusion, Physica A, 481, 276-283, (2017) [8] Zhou, Y.; Wang, J. R.; Zhang, L., Basic theory of fractional differential equations, (2016), World Scientific [9] Zhou, Y., Attractivity for fractional differential equations in Banach space, Appl. Math. Lett., 75, 1-6, (2018) · Zbl 1380.34025 [10] Zou, G., Galerkin finite element method for time-fractional stochastic diffusion equations, Comput. Appl. Math., (2018) · Zbl 1432.65153 [11] Zou, G.; Lv, G.; Wu, J. L., On the regularity of weak solutions to space-time fractional stochastic heat equations, Statist. Probab. Lett., 139, 84-89, (2018) · Zbl 1398.60080 [12] Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268, 4, 298-305, (2000) · Zbl 0948.81595 [13] Laskin, N., Fractional quantum mechanics, Phys. Rev. E, 62, 3135-3145, (2000) [14] Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66, (2002) [15] Guo, B.; Huang, D., Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys., 53, (2012) [16] Lions, P.-L., The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072, (1980) · Zbl 0453.47042 [17] Skupin, S.; Grech, M.; Królikowski, W., Rotating soliton solutions in nonlocal nonlinear media, Opt. Lett., 16, 9118-9131, (2008) [18] Shen, M.; Zheng, J.; Kong, Q.; Lin, Y.; Jeng, C.; Lee, R.; Krolikowski, W., Stabilization of counter-rotating vortex pairs in nonlocal media, Phys. Rev. A, 86, (2012) [19] Lieb, E. H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118, 2, 349-374, (1983) · Zbl 0527.42011 [20] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 5, 521-573, (2012) · Zbl 1252.46023 [21] Servadei, R.; Valdinoci, E., Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389, 887-898, (2012) · Zbl 1234.35291 [22] Ackermann, N., On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248, 423-443, (2004) · Zbl 1059.35037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.