Bifurcation of traveling spot patterns in 2-dimensional domain

研究了二维域上旅游斑点模式的分叉问题。我们的模型方程是包含非定域项的两组分反应扩散系统。。在数值上，它们呈现稳定地从站立光斑图案分叉出来的行进光斑图案。理论上，我们知道没有非定域项的二分量反应扩散系统不会出现这种现象。然后首先我们认为这种现象来自于非本地术语。但后来我们成功地将具有非局域项的二分量系统重写为不含非局域项的三分量反应扩散系统。我们认为，这种现象源于扩散比、反应比和非线性项之间的微妙平衡。这使我们能够对它们进行数学分析。在一维情形下，我们在双稳假设下证明了旅行正解和旅行后解的存在性和稳定性。现在，我们试图构建连接这些旅行前向和后向解决方案的站立脉冲解决方案。在稳定性方面，当参数改变时，驻留脉冲解首先在异相(非对称)模式下失稳，然后在同相(对称)模式下失稳。此时，前后解的稳定性信息以及这两种解的连接方式将有助于我们分析其稳定性。这些新的分支解是稳定的行波脉冲解。在二维情况下，数值模拟证实了这些系统具有稳定的二维光斑图案。对于这种情况，我们将证明径向对称驻留脉冲解的存在，然后捕捉旅行光斑图案作为径向对称驻留脉冲解的失稳。

Bifurcation problem of traveling spot patterns in 2-dimensional domain is studied. Our model equations are 2- component reaction-diffusion systems including a non-local term. . Numerically they exhibit traveling spot patterns which is bifurcated from standing spot patterns stably. Theoretically we know that 2-component reaction-diffusion systems without a non-local term do not exhibit such phenomena. Then first we considered that such phenomena came from the non-local term. But later we could succeed in rewriting the 2-component systems with the non-local term into the 3-component reaction-diffusion systems without a non-local term. We conclude that such phenomena come from the delicate balance of the ratio of diffusions, the ratio of reactions and the nonlinear terms. This enables us to analyze them mathematically. In 1-dimensional case, we showed the existence and the stability of traveling front and back solutions under the assumption of bistability. Now we try to construct standing pulse solutions connecting these traveling front and back solutions. For the stability property, we will show that standing pulse solutions destabilize under the out-of-phase (asymmetric) modes first and the in-phase (symmetric) modes secondly when some parameter is changed. At this time, the informatin of the stability of traveling front and back solutions and the connecting manner of these two solutions will help us to analyze the stability property. These new bifurcated solutions are stable traveling pulse solutions. In 2-dimensional case, it is confirmed numerically that these systems have stable 2-dim traveling spot patterns. For this case, we will show the existence of radial symmetric standing pulse solutions and then catch traveling spot patterns as the destabilization of radial symmetric standing pulse solutions.