Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (3): 784-806.

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Statistical Solutions and Its Limiting Behavior for the Impulsive Discrete Ginzburg-Landau Equations

Caidi Zhao1,*(),Huite Jiang1,Chunqiu Li1,Caraballo Tomás2   

  1. 1 Department of Mathematics, Wenzhou University, Zhejiang Wenzhou 325035
    2 Departmento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Mathmáticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain
  • Received:2021-04-23 Online:2022-06-26 Published:2022-05-09
  • Contact: Caidi Zhao
  • Supported by:
    the NSFC(11971356);the NSF of Zhejiang Province(LY17A010011)


In this article we first prove the global well-posedness of the impulsive discrete Ginzburg-Landau equations. Then we establish that the generated process by the solution operators possesses a pullback attractor and a family of invariant Borel probability measures. Further, we formulate the definition of statistical solution for the addressed impulsive system and prove the existence. Our results reveal that the statistical solution of the impulsive system satisfies merely the Liouville type theorem piecewise, which implies that the Liouville type equation for impulsive system will not always hold true on the interval containing any impulsive point. Finally, we prove that the statistical solution of the impulsive discrete Ginzburg-Landau equations converges to that of the impulsive discrete Schrödinger equations.

Key words: Statistical solution, Impulsive differential equation, Liouville type theorem, Discrete complex Ginzburg-Landau equation, Discrete Schrödinger equation

CLC Number: 

  • O175.8