Bifurcation of traveling spot patterns in 2-dimensional domain

研究了二维区域中的旅行斑图分岔问题。我们的模型方程是包含一个非局部项的两组分反应扩散方程组。.在数值上，它们表现出稳定地从静止点模式分叉的行进点模式。理论上我们知道，没有非局部项的二分量反应扩散方程组不存在这样的现象。然后我们首先考虑这种现象来自于非局部项。但后来我们可以成功地将带非局部项的二分量方程组改写为不带非局部项的三分量反应扩散方程组。我们的结论是，这种现象来自于扩散比，反应比和非线性项的微妙平衡。这使我们能够从数学上分析它们。在一维情形下，在双稳定性假设下，证明了行波解的存在性和稳定性。现在，我们尝试构造连接这些行进的前、后解的驻定脉冲解。对于稳定性性质，我们将表明，驻脉冲解不稳定下的相位（非对称）模式第一和相位（对称）模式，其次当一些参数被改变。此时，关于行波前、后解的稳定性信息以及这两个解的连接方式将有助于我们分析其稳定性。这些新的分叉解是稳定的行波脉冲解。在二维情况下，数值计算证实了这些系统具有稳定的二维行波斑图。在这种情况下，我们将证明径向对称驻脉冲解的存在性，然后将行进光斑图案作为径向对称驻脉冲解的不稳定性。

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.