133
Views
0
CrossRef citations to date
0
Altmetric
Articles

Time fractional and space nonlocal stochastic nonlinear Schrödinger equation driven by Gaussian white noise

, , &
Pages 123-136 | Received 13 May 2017, Accepted 05 Jun 2018, Published online: 05 Apr 2019

References

  • Bao, W., Cai, Y. (2012). Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Relat. Models 6(1):1–135.
  • Laskin, N. (2000). Fractional quantum mechanics. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(3 Pt A):3135–3145.
  • Laskin, N. (2000). Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4–6):298–305.
  • Guo, B., Han, Y., Xin, J. (2008). Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl. Math. Comput. 204(1):468–477.
  • Esquivel, L., Kaikina, E. (2016). Neumann problem for nonlinear Schrödinger equation with the Riesz fractional derivative operator. J. Differ. Equat. 260(7):5645–5677.
  • Li, S., Ding, Y., Chen, Y. (2015). Concentrating standing waves for the fractional Schrödinger equation with critical nonlinearities. Bound. Value Probl. 2015(1):1–26.
  • Guo, X., Xu, M. (2006). Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47(8):082104.
  • Shang, X., Zhang, J. (2015). Concentrating solutions of nonlinear fractional Schrödinger equation with potentials. J. Differ. Equ. 258(4):1106–1128.
  • Naber, M. (2004). Time fractional Schrödinger equation. J. Math. Phys. 45(8):3339–3352.
  • Iomin, A. (2009). Fractional-time quantum dynamics. Phys. Rev. E Stat. Nonlin. Soft Matter. Phys. 80(2 Pt 1):022103.
  • Iomin, A. (2011). Fractional-time Schrödinger equation: fractional dynamics on a comb. Chaos Solitons Fractals 44(4–5):348–452.
  • Dong, J. P., Xu, M. Y. (2008). Space–time fractional Schrödinger equation with time-independent potentials. J. Math Anal. Appl. 344(2):1005–1017.
  • Górka, P., Prado Trujillo, H. J. (2017). The time fractional Schrödinger equation on Hilbert space. Integr. Equ. Oper. Theory 87(1):1–14.
  • Bayin, S. S. (2013). Time fractional Schrödinger equation: Fox’s H-functions and the effective potential. J. Math. Phys. 54(1):012103.
  • Narahari Achar, B. N., Yale, B. T., Hanneken, J. W. (2013). Time fractional Schrödinger equation revisited. Adv. Math. Phys. 2013(1):290216.
  • Mohebbi, A., Abbaszadeh, M., Dehghan, M. (2013). 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.
  • Hong, B. J., Lu, D. C. (2014). Modified fractional variational iteration method for solving the generalized time–space fractional Schrödinger equation. Sci. World J. 2014:964643.
  • Wei, L. L., He, Y. N., Zhang, X. D., Wang, S. L. (2012). Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation. Finite Elem. Anal. Des. 59:28–34.
  • Herzallah, M. A. E., Gepreel, K. A. (2012). Approximate solution to the time–space fractional cubic nonlinear Schrödinger equation. Appl. Math. Mod. 36(11):5678–5685.
  • Falkovich, G. E., Kolokolov, I., Lebedev, V., Turitsyn, S. K. (2001). Statistics of soliton-bearing systems with additive noise. Phys. Rev. E 63(2):235–241.
  • Bang, O., Christiansen, P. L., If, F., Rasmussen, K. O., Gaididei, Y. B. (1995). White noise in the two-dimensional nonlinear Schrödinger equation. J. Appl. Anal. 57(1–2):3–15.
  • Belaouar, R., de Bouard, A., Debussche, A. (2015). The stochastic nonlinear Schrödinger equation in H1. Stoch Pde. Anal. Comp. 3(1):103.
  • Da Prato, G., Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge University Press.
  • Ye, H., Gao, J., Ding, Y. (2007). A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Appl. Anal. 328(2):1075–1081.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.