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Time fractional and space nonlocal stochastic nonlinear Schrödinger equation driven by Gaussian white noise. (English) Zbl 1416.35242

Summary: We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Ulenbeck equations by the generalized Mittag-Leffler functions and Mainardi function, and establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic nonlinear Schrödinger equation driven by Gaussian white noise. In addition, the global mild solution is also shown.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R11 Fractional partial differential equations
35B65 Smoothness and regularity of solutions to PDEs
35Q82 PDEs in connection with statistical mechanics
33E12 Mittag-Leffler functions and generalizations
35R60 PDEs with randomness, stochastic partial differential equations
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[1] Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1, 1-135 (2012) · Zbl 1266.82009
[2] Laskin, N., Fractional quantum mechanics, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics, 62, 3, 3135-3145 (2000)
[3] Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268, 4-6, 298-305 (2000) · Zbl 0948.81595
[4] Guo, B.; Han, Y.; Xin, J., Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput, 204, 1, 468-477 (2008) · Zbl 1163.35483
[5] Esquivel, L.; Kaikina, E., Neumann problem for nonlinear Schrödinger equation with the Riesz fractional derivative operator, J. Differ. Equat, 260, 7, 5645-5677 (2016) · Zbl 1337.35133
[6] Li, S.; Ding, Y.; Chen, Y., Concentrating standing waves for the fractional Schrödinger equation with critical nonlinearities, Bound. Value Probl, 2015, 1, 1-26 (2015) · Zbl 1333.35223
[7] Guo, X.; Xu, M., Some physical applications of fractional Schrödinger equation, J. Math. Phys, 47, 8, 082104 (2006) · Zbl 1112.81028
[8] Shang, X.; Zhang, J., Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differ. Equ, 258, 4, 1106-1128 (2015) · Zbl 1348.35243
[9] Naber, M., Time fractional Schrödinger equation, J. Math. Phys, 45, 8, 3339-3352 (2004) · Zbl 1071.81035
[10] Iomin, A., Fractional-time quantum dynamics, Phys. Rev. E Stat. Nonlin. Soft Matter. Phys, 80, 2-1, 022103 (2009)
[11] Iomin, A., Fractional-time Schrödinger equation: fractional dynamics on a comb, Chaos Solitons Fractals, 44, 4-5, 348-452 (2011) · Zbl 1225.81053
[12] Dong, J. P.; Xu, M. Y., Space-time fractional Schrödinger equation with time-independent potentials, J. Math Anal. Appl, 344, 2, 1005-1017 (2008) · Zbl 1140.81357
[13] Górka, P.; Prado Trujillo, H. J., The time fractional Schrödinger equation on Hilbert space, Integr. Equ. Oper. Theory, 87, 1, 1-14 (2017) · Zbl 1368.35231
[14] Bayin, S. S., Time fractional Schrödinger equation: Fox’s H-functions and the effective potential, J. Math. Phys, 54, 1, 012103 (2013) · Zbl 1280.81034
[15] Narahari Achar, B. N.; Yale, B. T.; Hanneken, J. W., Time fractional Schrödinger equation revisited, Adv. Math. Phys, 2013, 1, 290216 (2013) · Zbl 1292.81031
[16] Mohebbi, A.; Abbaszadeh, M.; Dehghan, M., The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics, Engrg. Anal. Bound. Elem, 37, 2, 475-485 (2013) · Zbl 1352.65397
[17] Hong, B. J.; Lu, D. C., Modified fractional variational iteration method for solving the generalized time-space fractional Schrödinger equation, Sci. World J, 2014, 964643 (2014)
[18] Wei, L. L.; He, Y. N.; Zhang, X. D.; Wang, S. L., Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation, Finite Elem. Anal. Des, 59, 28-34 (2012)
[19] Herzallah, M. A. E.; Gepreel, K. A., Approximate solution to the time-space fractional cubic nonlinear Schrödinger equation, Appl. Math. Mod, 36, 11, 5678-5685 (2012) · Zbl 1254.65115
[20] Falkovich, G. E.; Kolokolov, I.; Lebedev, V.; Turitsyn, S. K., Statistics of soliton-bearing systems with additive noise, Phys. Rev. E, 63, 2, 235-241 (2001)
[21] Bang, O.; Christiansen, P. L.; If, F.; Rasmussen, K. O.; Gaididei, Y. B., White noise in the two-dimensional nonlinear Schrödinger equation, J. Appl. Anal, 57, 1-2, 3-15 (1995) · Zbl 0842.35111
[22] Belaouar, R.; De Bouard, A.; Debussche, A., The stochastic nonlinear Schrödinger equation in \(H^1\), Stoch Pde. Anal. Comp, 3, 1, 103 (2015) · Zbl 1314.35159
[23] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (1992), Cambridge: Cambridge University Press, Cambridge · Zbl 0761.60052
[24] Ye, H.; Gao, J.; Ding, Y., A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Appl. Anal, 328, 2, 1075-1081 (2007) · Zbl 1120.26003
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