Lattice Boltzmann Method For Nonlinear Waves In Excitable Media

The characteristic of nonlinear waves in excitable media is one of the current research focus. The spiral wave and its fragmentation usually do harm to the practical systems. Therefore, it has greatly practical significance to study the characteristic of the nonlinear wave. With many merits such as higher precision, simple code and excellent stability, Lattice Boltzmann method has become an effective tool for studying the nature of nonlinear wave.In this paper, we studied the nonlinear wave in the Selkov reaction-diffusion system by using Lattice Boltzmann method. First of all, we established the 9-speed square lattice model for the reaction-diffusion equations and deduced the reaction-diffusion equations from the Lattice Boltzmann equation. Next, we divided the two-dimensional system into 300×300 grids and simulated it under the no-flow boundary condition by using computer. The main works of this paper include three parts as follow:First, when taking the parameters as a =0.76, b =0.02,χ=0.1,κ=5.0, D_X = D_Y=0.1, the numerical simulation results show that the system is excitable. Under the same parameters, the system can evolve to different nonlinear waves starting from different initial states, and the system can evolve into three different states such as the spiral wave state, spatiotemporal chaos state and uniform state starting from the same state with different parameters. It is also found that the spiral wave instability in the Selkov reaction system is the Doppler instability. The system phase diagrams within a certain range of parameters are shown through the simulation of the system evolution under a variety of parameters.Second, based on the theory of the lattice Boltzmann method, it is defined that the internal energy function is .Numerical simulation results show these as follows. In the homogeneous state, the internal energy of excitable medium increases linearly with the bifurcation parameter a increases, while in the spiral waves, the internal energy of excitable medium decreases in the index form with the bifurcation parameter a increases. Under the same parameter and the states of the system as traveling waves, target waves, spiral waves, respectively, internal energy of the system changes slightly and periodically with time. It is found that the mean internal energy in the system have little discrepancy, the reason is that the same parameters mean that the energy of the system is closed, even for the different stable states of waves. The calculation of internal energy of system before and after the spiral wave stability has been done, and the results show that internal energy of the system decreases rapidly when it loses stability, which indicates that spiral wave instability due to the poor support of the energy of the system.Third, basing on the nine-velocity grid model, it is defined that an entropy function according to the Lattice Boltzmann method as Simulations of the entropy of the different wave states with the same parameters have been done, and the entropy of the spiral wave before and after it loses instability is shown in this text, too. Under the same parameters,the entropy of different stable wave states changes lightly and periodically with time. After the treatment, it is found that entropy is not the same for different wave states, which indicates that the order degree of each wave states is not the same. When spiral wave became instability, the entropy increases suddenly, which indicates that the system evolve to the state whose entropy is bigger when the system undergoing the process of spontaneous phase transition.As the complexity of nonlinear systems, many issues about the nature of nonlinear waves in excitable media need further research. At the end of this paper, the summary and outlook are shown.