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Statistical Solution for the Nonlocal Discrete Nonlinear Schrödinger Equation

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Abstract

In this article, we consider the nonlocal discrete nonlinear Schrödinger equation. We first prove that the associated process has a pullback-\({{\mathcal {D}}}_\delta \) attractor by overcoming the difficulties caused by the nonlocal operator. Then we establish the existence of a unique family of invariant Borel probability measures carried by the pullback attractor. Finally, we further construct statistical solutions for this nonlocal equation on infinite lattices.

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References

  1. Alfimov, G.L., Kevrekidis, P.G., Konotop, V.V., Salerno, M.: Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential. Phys. Rev. E 66, 046608 (2002)

    MathSciNet  Google Scholar 

  2. Bates, P.W., Zhang, C.: Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete Contin. Dyn. Syst. 16(1), 253–277 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Bronzi, A.C., Mondaini, C.F., Rosa, R.M.S.: Trajectory statistical solutions for three-dimensional Navier–Stokes-like systems. SIAM J. Math. Anal. 46(3), 1893–1921 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Bronzi, A.C., Mondaini, C.F., Rosa, R.M.S.: Abstract framework for the theory of statistical solutions. J. Differ. Equ. 260, 8428–8484 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Chekroun, M., Glatt-Holtz, N.E.: Invariant measures for dissipative dynamical systems: abstract results and applications. Commun. Math. Phys. 316, 723–761 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Caraballo, T., Kloeden, P.E., Real, J.: Invariant measures and statistical solutions of the globally modified Navier–Stokes equations. Discrete Contin. Dyn. Syst.-B 10, 761–781 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Chebab, J.P., Dumont, S., Goubet, O., Moatassime, H., Abounouh, M.: Discrete Schrödinger equation and dissipative dynamical systems. Commun. Pure Appl. Anal. 7(2), 211–227 (2006)

    MATH  Google Scholar 

  8. Chen, T., Zhou, S., Zhao, C.: Attractors for discrete nonlinear Schrödinger equation with delay. Acta Math. Appl. Sin. Engl. Ser. 26(4), 633–642 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Erneux, T., Nicolis, G.: Propagating waves in discrete bistable reaction diffusion systems. Phys. D 67, 237–244 (1993)

    MathSciNet  MATH  Google Scholar 

  10. Foias, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  11. Hennig, D.: Existence and congruence of global attractors for damped and forced integrable and 458 nonintegrable discrete nonlinear Schrödinger equations. J. Dyn. Differ. Equ. 1–19 (2021). https://doi.org/10.1007/s10884-021-10104-3

  12. Han, X., Kloeden, P.E.: Non-autonomous lattice systems with switching effects and delayed recovery. J. Differ. Equ. 261(6), 2986–3009 (2016)

    MathSciNet  MATH  Google Scholar 

  13. He, Y., Li, C.Q., Wang, J.T.: Invariant measures and statistical solutions for the nonautonomous discrete modified Swift–Hohenberg equation. Bull. Malays. Math. Sci. Soc. 44, 3819–3837 (2021)

    MathSciNet  MATH  Google Scholar 

  14. Ignat, I.L., Rossi, J.D.: Asymptotic behaviour for a nonlocal diffusion equation on a lattice. Z. Angew. Math. Phys. 59(5), 918–925 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Kevrekidis, P.G.: The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, vol. 232. Springer (2009)

  16. Keener, J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572 (1987)

    MathSciNet  MATH  Google Scholar 

  17. Kevrekidis, P.G., Rasmussen, K., Bishop, A.R.: The discrete nonlinear Schrödinger equation: a survey of recent results. Int. J. Mod. Phys. B 15(21), 2833–2900 (2001)

    Google Scholar 

  18. Kloeden, P.E., Marín-Rubio, P., Real, J.: Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier–Stokes equations. Commun. Pure Appl. Anal. 8, 785–802 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Kirkpatrick, K., Lenzmann, E., Staffilani, G.: On the continuum limit for discrete NLS with long-range lattice interactions. Commun. Math. Phys. 317, 563–591 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Karachalios, N.I., Yannacopoulos, A.N.: Global existence and compact attractors for the discrete nonlinear Schrödinger equation. J. Differ. Equ. 217(1), 88–123 (2005)

    MATH  Google Scholar 

  21. Łukaszewicz, G.: Pullback attractors and statistical solutions for 2-D Navier–Stokes equations. Discrete Contin. Dyn. Syst.-B 9, 643–659 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Li, C.Q., Wang, J.T.: On the forward dynamical behaviour of nonautonomous lattice dynamical systems. J. Differ. Equ. Appl. 27, 1052–1080 (2021)

    MathSciNet  MATH  Google Scholar 

  23. Li, C.C., Li, C.Q., Wang, J.T.: Statistical solution and Liouville type theorem for coupled Schrödinger–Boussinesq equations on infinite lattices. Discrete Contin. Dyn. Syst.-B 27(10), 6173–6196 (2022)

    MathSciNet  MATH  Google Scholar 

  24. Łukaszewicz, G., Robinson, J.C.: Invariant measures for nonautonomous dissipative dynamical systems. Discrete Contin. Dyn. Syst. 34(10), 4211–4222 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Łukaszewicz, G., Real, J., Robinson, J.C.: Invariant measures for dissipative dynamical systems and generalised Banach limits. J. Dyn. Differ. Equ. 23, 225–250 (2011)

    MATH  Google Scholar 

  26. Laskin, N., Zaslavsky, G.: Nonlinear fractional dynamics on a lattice with long range interactions. Physica A 368(1), 38–54 (2006)

    Google Scholar 

  27. Morsch, O., Oberthaler, M.: Dynamics of Bose–Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179 (2006)

    Google Scholar 

  28. Mingaleev, S.F., Christiansen, P.L., Gaididei, Yu.B., Johansson, M., Rasmussen, K.: Models for energy and charge transport and storage in biomolecules. J. Biol. Phys. 25, 41–63 (1999)

    Google Scholar 

  29. Peyrard, M.: Nonlinear dynamics and statistical physics of DNA. Nonlinearity 17, R1 (2004)

    MathSciNet  MATH  Google Scholar 

  30. Pereira, J.M.: Global attractor for a generalized discrete nonlinear Schrödinger equation. Acta Appl. Math. 134(1), 173–183 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Pereira, J.M.: Pullback attractor for a nonlocal discrete nonlinear Schrödinger equation with delays. Electron. J. Qual. Theory Differ. Equ. 93, 1–18 (2021)

    Google Scholar 

  32. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    MathSciNet  MATH  Google Scholar 

  33. Pacciani, P., Konotop, V.V., Perla Menzala, G.: On localized solutions of discrete nonlinear Schrödinger equation. An exact result. Phys. D. 204(1–2), 122–133 (2005)

    MathSciNet  MATH  Google Scholar 

  34. Vekslerchik, V.E., Konotop, V.V.: Discrete nonlinear Schrödinger equation under nonvanishing boundary conditions. Inverse Prob. 8(6), 889 (1992)

    MATH  Google Scholar 

  35. Wang, X.: Upper-semicontinuity of stationary statistical properties of dissipative systems. Discrete Contin. Dyn. Syst. 23, 521–540 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Wu, S., Huang, J.H.: Invariant measure and statistical solutions for nonautonomous discrete Klein–Gordon–Schrödinger type equations. J. Appl. Anal. Comput. 10(4), 1516–1533 (2020)

    MathSciNet  MATH  Google Scholar 

  37. Wang, J.T., Zhang, X., Zhao, C.: Statistical solutions for a nonautonomous modified Swift–Hohenberg equation. Math. Methods Appl. Sci. 44, 14502–14516 (2021)

    MathSciNet  MATH  Google Scholar 

  38. Wang, C., Xue, G., Zhao, C.: Invariant Borel probability measures for discrete long-wave-short-wave resonance equations. Appl. Math. Comput. 339, 853–865 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Wang, J.T., Zhao, C., Caraballo, T.: Invariant measures for the 3D globally modified Navier–Stokes equations with unbounded variable delays. Commun. Nonlinear Sci. Numer. Simul. 91, 105459 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Zhao, C., Yang, L.: Pullback attractor and invariant measures for the globally modified Navier–Stokes equations. Commun. Math. Sci 15(6), 1565–1580 (2017)

    MathSciNet  MATH  Google Scholar 

  41. Zhao, C., Xue, G., Łukaszewicz, G.: Pullback attractors and invariant measures for discrete Klein–Gordon–Schrödinger equations. Discrete Contin. Dyn. Syst.-B 23(9), 4021–4044 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Zhao, C., Caraballo, T.: Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier–Stokes equations. J. Differ. Equ. 266, 7205–7229 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Zhao, C., Li, Y., Łukaszewicz, G.: Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids. Z. Angew. Math. Phys. 71, 1–24 (2020)

    MathSciNet  MATH  Google Scholar 

  44. Zhao, C., Caraballo, T., Łukaszewicz, G.: Statistical solution and Liouville type theorem for the Klein–Gordon–Schrödinger equations. J. Differ. Equ. 281, 1–32 (2021)

    MATH  Google Scholar 

  45. Zhu, Z., Sang, Y., Zhao, C.: Pullback attractor and invariant measures for the discrete Zakharov equations. J. Appl. Anal. Comput. 9, 2333–2357 (2019)

    MathSciNet  MATH  Google Scholar 

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Funding

This work is supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ22A010002 and by the Foundation of Department of Education of Zhejiang Province under Grant Y202248858.

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Authors and Affiliations

  1. College of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou, 325035, Zhejiang, China

    Congcong Li

  2. Department of Mathematics, Wenzhou University, Wenzhou, 325035, Zhejiang, People’s Republic of China

    Chunqiu Li

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  1. Congcong Li
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Correspondence to Chunqiu Li.

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Communicated by Rosihan M. Ali.

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Li, C., Li, C. Statistical Solution for the Nonlocal Discrete Nonlinear Schrödinger Equation. Bull. Malays. Math. Sci. Soc. 46, 106 (2023). https://doi.org/10.1007/s40840-023-01508-z

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