Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise

References

• Afraimovich, V. S., Nekorkin, V. I. (1994). Chaos of traveling waves in a discrete chain of diffusively coupled maps. Int. J. Bifurcation Chaos. 4(3):631–637. DOI: 10.1142/S0218127494000459.
   Web of Science ®Google Scholar
• Arnold, L. (1974). Stochastic Differential Equations: Theory and Applications. New York, NY: John Wiley & Sons Inc.
   Google Scholar
• Bates, P. W., Chen, X., Chmaj, A. (2003). Traveling waves of bistable dynamics on a lattice. SIAM J. Math. Anal. 35(2):520–546. DOI: 10.1137/S0036141000374002.
   Web of Science ®Google Scholar
• Bates, P. W., Chmaj, A. (1999). On a discrete convolution model for phase transitions. Arch. Ration. Mech. Anal. 150(4):281–305. DOI: 10.1007/s002050050189.
   Web of Science ®Google Scholar
• Bates, P. W., Lu, K., Wang, B. (2001). Attractors for lattice dynamical systems. Int. J. Bifur. Chaos. 11(1):143–153. DOI: 10.1142/S0218127401002031.
   Web of Science ®Google Scholar
• Bates, P. W., Lisei, H., Lu, K. (2006). Attractors for stochastic lattice dynamical systems. Stoch. Dyn. 6(1):1–21. DOI: 10.1142/S0219493706001621.
   Web of Science ®Google Scholar
• Bates, P. W., Lu, K., Wang, B. (2009). Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differ. Equat. 246(2):845–869. DOI: 10.1016/j.jde.2008.05.017.
   Web of Science ®Google Scholar
• Bates, P. W., Lu, K., Wang, B. (2014). Attractors of non-autonomous stochastic lattice systems in weighted spaces. Physica D. 289:32–50. DOI: 10.1016/j.physd.2014.08.004.
   Web of Science ®Google Scholar
• Bell, J., Cosner, C. (1984). Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons. Quart. Appl. Math. 42(1):1–14. DOI: 10.1090/qam/736501.
   Web of Science ®Google Scholar
• Beyn, W. J., Pilyugin, S. Y. (2003). Attractors of reaction diffusion systems on infinite lattices. J. Dyn. Differ. Equat. 15(2/3):485–515. DOI: 10.1023/B:JODY.0000009745.41889.30.
   Google Scholar
• Brzezniak, Z., Ondrejat, M., Seidler, J. (2016). Invariant measures for stochastic nonlinear beam and wave equations. J. Differ. Equat. 260:4157–4179. DOI: 10.1016/j.jde.2015.11.007.
   Web of Science ®Google Scholar
• Brzezniak, Z., Motyl, E., Ondrejat, M. (2017). Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains. Ann. Probab. 45:3145–3201. DOI: 10.1214/16-AOP1133.
   Web of Science ®Google Scholar
• Caraballo, T., Langa, J. A. (2003). On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems. Dyn. Cont. Discrete Impuls. Syst. Series A: Math. Anal. 10:491–513.
   Google Scholar
• Caraballo, T., Garrido-Atienza, M. J., Schmalfuss, B., Valero, J. (2008). Non-autonomous and random attractors for delay random semilinear equations without uniqueness. DCDS-A. 21:415–443. DOI: 10.3934/dcds.2008.21.415.
   Web of Science ®Google Scholar
• Caraballo, T., Garrido-Atienza, M. J., Schmalfuss, B., Valero, J. (2010). Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. DCDS-B. 14(2):439–455. DOI: 10.3934/dcdsb.2010.14.439.
   Web of Science ®Google Scholar
• Caraballo, T., Lu, K. (2008). Attractors for stochastic lattice dynamical systems with a multiplicative noise. Front. Math. China 3(3):317–335. DOI: 10.1007/s11464-008-0028-7.
   Web of Science ®Google Scholar
• Caraballo, T., Morillas, F., Valero, J. (2012). Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity. J. Differ. Equat. 253(2):667–693. DOI: 10.1016/j.jde.2012.03.020.
   Web of Science ®Google Scholar
• Chow, S. N., Mallet-Paret, J. (1995). Pattern formation and spatial chaos in lattice dynamical systems, I, II. IEEE Trans. Circuits Syst. I. 42(10):746–751. DOI: 10.1109/81.473583.
   Google Scholar
• Chow, S. N., Mallet-Paret, J., Shen, W. (1998). Traveling waves in lattice dynamical systems. J. Differ. Equat. 49:248–291. DOI: 10.1006/jdeq.1998.3478.
   Google Scholar
• Chow, S. N., Mallet-Paret, J., Van Vleck, E. S. (1996). Pattern formation and spatial chaos in spatially discrete evolution equations. Random Comput. Dyn. 4:109–178.
   Google Scholar
• Chow, S. N., Shen, W. (1995). Dynamics in a discrete Nagumo equation: Spatial topological chaos. SIAM J. Appl. Math. 55(6):1764–1781. DOI: 10.1137/S0036139994261757.
   Web of Science ®Google Scholar
• Chua, L. O., Roska, T. (1993). The CNN paradigm. IEEE Trans. Circuits Syst. I. 40(3):147–156. DOI: 10.1109/81.222795.
   Google Scholar
• Chua, L. O., Yang, Y. (1988). Cellular neural networks: Theory. IEEE Trans. Circuits Syst. 35(10):1257–1272. DOI: 10.1109/31.7600.
   Google Scholar
• Chueshov, I., Scheutzow, M. (2004). On the structure of attractors and invariant measures for a class of monotone random systems. Dyn. Syst. 19(2):127–144. DOI: 10.1080/1468936042000207792.
   Web of Science ®Google Scholar
• Crauel, H., Flandoli, F. (1994). Attractors for random dynamical systems. Probab. Th. Rel. Fields 100(3):365–393. DOI: 10.1007/BF01193705.
   Web of Science ®Google Scholar
• Crauel, H., Debussche, A., Flandoli, F. (1997). Random attractors. J. Dyn. Differ. Equat. 9(2):307–341. DOI: 10.1007/BF02219225.
   Google Scholar
• Eckmann, J., Hairer, M. (2001). Invariant measures for stochastic partial differential equations in unbounded domains. Nonlinearity 14(1):133–151. DOI: 10.1088/0951-7715/14/1/308.
   Web of Science ®Google Scholar
• Ekren, I., Kukavica, I., Ziane, M. (2017). Existence of invariant measures for the stochastic damped Schrödinger equation. Stoch. PDE: Anal. Comp. 5(3):343–367. DOI: 10.1007/s40072-016-0090-1.
   Google Scholar
• Elmer, C. E., Van Vleck, E. S. (1999). Analysis and computation of traveling wave solutions of bistable differential-difference equations. Nonlinearity 12(4):771–798. DOI: 10.1088/0951-7715/12/4/303.
   Web of Science ®Google Scholar
• Elmer, C. E., Van Vleck, E. S. (2001). Traveling waves solutions for bistable differential-difference equations with periodic diffusion. SIAM J. Appl. Math. 61:1648–1679. DOI: 10.1137/S0036139999357113.
   Web of Science ®Google Scholar
• Erneux, T., Nicolis, G. (1993). Propagating waves in discrete bistable reaction diffusion systems. Physica D 67(1–3):237–244. DOI: 10.1016/0167-2789(93)90208-I.
   Web of Science ®Google Scholar
• Flandoli, F., Schmalfuss, B. (1996). Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stoch. Stoch. Rep. 59(1–2):21–45. DOI: 10.1080/17442509608834083.
   Google Scholar
• Garrido-Atienza, M. J., Ogrowsky, A., Schmalfuss, B. (2011). Random differential equations with random delays. Stoch. Dyn. 11(02n03):369–388. DOI: 10.1142/S0219493711003358.
   Google Scholar
• Han, X. (2011). Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise. J. Math. Anal. 376(2):481–493. DOI: 10.1016/j.jmaa.2010.11.032.
   Web of Science ®Google Scholar
• Han, X., Shen, W., Zhou, S. (2011). Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Differ. Equat. 250(3):1235–1266. DOI: 10.1016/j.jde.2010.10.018.
   Web of Science ®Google Scholar
• Huang, J., Shen, W. (2009). Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete Cont. Dyn. Syst. 24:855–882.
   Web of Science ®Google Scholar
• Kapval, R. (1991). Discrete models for chemically reacting systems. J. Math. Chem. 6:113–163.
   Web of Science ®Google Scholar
• Karachalios, N. I., Yannacopoulos, A. N. (2005). Global existence and compact attractors for the discrete nonlinear Schrodinger equation. J. Differ. Equat. 217(1):88–123. DOI: 10.1016/j.jde.2005.06.002.
   Web of Science ®Google Scholar
• Keener, J. P. (1987). Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47(3):556–572. DOI: 10.1137/0147038.
   Web of Science ®Google Scholar
• Keener, J. P. (1991). The effects of discrete gap junction coupling on propagation in myocardium. J. Theor. Biol. 148(1):49–82. DOI: 10.1016/S0022-5193(05)80465-5.
   PubMed Web of Science ®Google Scholar
• Kim, J. (2006). On the stochastic Burgers equation with polynomial nonlinearity in the real line. DCDS-B. 6(4):835–866. DOI: 10.3934/dcdsb.2006.6.835.
   Web of Science ®Google Scholar
• Kim, J. (2006). On the stochastic Benjamin-Ono equation. J. Differ. Equat. 228(2):737–768. DOI: 10.1016/j.jde.2005.11.005.
   Web of Science ®Google Scholar
• Kim, J. (2004). Periodic and invariant measures for stochastic wave equations. Electr. J. Differ. Equat. 2004(5):1–30.
   Google Scholar
• Kim, J. (2006). Invariant measures for a stochastic nonlinear schrodinger equation. Indiana Univ. Math. J. 55(2):687–717. DOI: 10.1512/iumj.2006.55.2701.
   Web of Science ®Google Scholar
• Kloeden, P. E., Langa, J. A. (2007). Flattening, squeezing and the existence of random attractors. Proc. R Soc. A. 463(2077):163–181. DOI: 10.1098/rspa.2006.1753.
   Google Scholar
• Kloeden, P. E., Lorenz, T. (2012). Mean-square random dynamical systems. J. Differ. Equat. 253(5):1422–1438. DOI: 10.1016/j.jde.2012.05.016.
   Web of Science ®Google Scholar
• Langa, J. A., Suarez, A. (2002). Pullback permanence for non-autonomous partial differential equations. Electr. J. Differ. Equat. 2002(72):1–20. No.
   Google Scholar
• Lv, Y., Sun, J. (2006). Dynamical behavior for stochastic lattice systems. Chaos Solitons Fractals 27(4):1080–1090. DOI: 10.1016/j.chaos.2005.04.089.
   Web of Science ®Google Scholar
• Morillas, F., Valero, J. (2009). A Peano’s theorem and attractors for lattice dynamical systems. Int. J. Bifurcation Chaos 19(2):557–578. DOI: 10.1142/S0218127409023196.
   Web of Science ®Google Scholar
• Misiats, O., Stanzhytskyi, O., Yip, N. (2016). Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains. J. Theor. Probab. 29(3):996–1026. DOI: 10.1007/s10959-015-0606-z.
   Web of Science ®Google Scholar
• Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. New York, NY: Academic Press.
   Google Scholar
• Schmalfuss, B. (1992). Backward cocycles and attractors of stochastic differential equations. International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185–192.
   Google Scholar
• Wang, B. (2013). Random attractors for non-autonomous stochastic wave equations with multiplicative noise. DCDS-A. 34(1):269–300. DOI: 10.3934/dcds.2014.34.269.
   Web of Science ®Google Scholar
• Wang, B. (2006). Dynamics of systems on infinite lattices. J. Differ. Equat. 221(1):224–245. DOI: 10.1016/j.jde.2005.01.003.
   Web of Science ®Google Scholar
• Wang, B. (2019). Weak pullback attractors for mean random dynamical systems in Bochner spaces. J. Dyn. Differ. Equat. DOI: 10.1007/s10884-018-9696-5.
   Web of Science ®Google Scholar
• Wang, B. (2019). Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise. J. Math. Anal. Appl. 477(1):104–132. DOI: 10.1016/j.jmaa.2019.04.015.
   Web of Science ®Google Scholar
• Zinner, B. (1992). Existence of traveling wavefront solutions for the discrete nagumo equation. J. Differ. Equat. 96(1):1–27. DOI: 10.1016/0022-0396(92)90142-A.
   Web of Science ®Google Scholar