Acta mathematica scientia,Series A ›› 2019, Vol. 39 ›› Issue (5): 1170-1182.

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Gradient Flow Method in Nonlinear Optical Lattices

Ruifeng Zhang1,*(),Nan Liu2()   

  1. 1 College of Mathematics and Statistics, Henan University, Henan Kaifeng 475004
    2 Institute of Applied Physics and Computational Mathematics, Beijing 100088
  • Received:2018-05-04 Online:2019-10-26 Published:2019-11-08
  • Contact: Ruifeng Zhang;
  • Supported by:
    the NSFC(11471099);the NSFC(11671120)


In this paper, we study the existence of the steady state solutions for a classical Schrödinger equation in nonlinear optical lattices by means of gradient flow method. We first establish the existence of a global solution of the governing parabolic equation. Then we prove the convergence of the global solution to an equilibrium (i.e., a steady state solution in optical lattices model) as time goes to infinity. Furthermore, we provide an estimate on the convergence rate by using the Łojasiewicz-Simon inequality.

Key words: Gradient flow method, Schrödinger Equation, Łojasiewicz-Simon inequality, Steady state solutions

CLC Number: 

  • O175.26