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2688-1594

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## Electronic Research Archive

### Special issue on infinite dimensional dynamical systems and applications

Xiaoying Han, Auburn University, Auburn, AL, USA xzh0003@auburn.edu

Wan-Tong Li, Lanzhou University, Chinawtli@lzu.edu.cn

Zhi-Cheng Wang, School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, Chinawangzhch@lzu.edu.cn

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**Abstract:**

The asymptotic behaviour of an autonomous neural field lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice system is reformulated as an infinite dimensional ordinary delay differential equation on weighted sequence state space

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In this paper, two properties of the pullback attractor for a 2D non-autonomous micropolar fluid flows with delay on unbounded domains are investigated. First, we establish the

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In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary

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We study the random dynamics for the stochastic non-autonomous Kuramoto-Sivashinsky equation in the possibly non-dissipative case. We first prove the existence of a pullback attractor in the Lebesgue space of odd functions, then show that the fiber of the odd pullback attractor semi-converges to a nonempty compact set as the time-parameter goes to minus infinity and finally prove the measurability of the attractor. In a word, we obtain a longtime stable random attractor formed from odd functions. A key tool is the existence of a bridge function between Lebesgue and Sobolev spaces of odd functions.

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In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

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**Abstract:**

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

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**Abstract:**

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

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