Numerical approach for bifurcation of nonlinear problem

它试图再次弥补传统水波分岔研究中存在的一些漏洞。因此，在本研究中，对旋转流动进行数值模拟得到如下结果：我们的目标是在传统研究成果的基础上计算分支分支来改变涡度分布，并对极端波剖面进行分类。用Zeidler的方法计算了一个有涡量的自由边界问题。在进行数值实验的过程中，这种方法表明，当出现封闭涡流时，有时无法计算。直到现在，它理解存在可能是在恒定涡度的情况下。但在一般涡度分布的情况下，是否存在闭合涡是一个有待确认的问题。用有限差分法进行了数值模拟，得到了一些结果。至于深度，它只处理了有限的……更多的情况。我们的研究结果如下：下面，我们考虑涡度函数处处为正（或负）且随深度衰减的情况。而当表面张力作用时，其涡度分布的极值波型均为悬垂型或重叠型，即光滑边界。对于重力波，当涡度函数随水深衰减时，成为具有角点或尖点的极值波廓线。另一方面，当重力波的涡度恒定时，在到达极值波的途中会出现悬垂型波，极值波最后有一个拐角。根据我们的实验，只有在恒定涡度的情况下才能看到这种悬垂型的重力波。这是一个值得注意的引力波现象，需要在未来进行更多的研究。我们在台北中研院数学研究所举行的“台湾-日本非线性分析与应用数学联合会议”上发表了上述结果。在整理了几份数值数据后，拟将其整理成论文。少

It attempted to take up some problems that there was a loose end in the conventional research about the bifurcation of water waves once again. As a result in this research, it won a result like the following in case of numerical simulation of the rotational flow.Our aim is computing bifurcation branches for changing vorticity distribution and classifying extreme wave profiles along with our conventional research results. It used the way according to Zeidler to compute a free boundary problem with vorticity. In the process of proceeding with the numerical experiment, this way showed that it wasn't sometimes possible to compute when a closed eddy occurs. Until the present, it understands the existence might be in the case of constant vorticity. However it is a problem to be confirmed whether there occur a closed eddy in the case of general vorticity distribution. It did a numerical simulation by a finite difference method and it got some results. As for the depth, it handled only a finit … More e case.Our results in the research are as follows. Hereinafter, we considered the case where the vorticity function is positive (or negative) everywhere and decays with the depth. When the surface tension works, however it gives vorticity distribution, all the extreme wave profiles are overhanging type or overlapped type, namely smooth boundary. For gravity waves, when the vorticity function decays with the water depth, it becomes the extreme wave profile has corner or cusp. On the other hand, in the case of the constant vorticity of gravity waves, there appears an overhanging type of waves on the way to the extreme wave, which has a corner at last. According to our experiments, it is only in the case of the constant vorticity to be seen such overhanging type of gravity waves. It is remarkable phenomenon for gravity waves and it needs to be more reviewed by it in the future.We presented the above results in "Taiwan-Japan Joint Conference on Nonlinear Analysis and Applied Mathematics" at Institute of Mathematics Academia Sinica of Taipei. It plans to gather as the paper after arranging for a few pieces of numerical data. Less