Numerical approach for bifurcation of nonlinear problem

再次尝试解决常规水波分叉研究中存在的一些松散问题。结果表明，在数值模拟旋转流的情况下，我们的目标是计算改变涡度分布的分叉分支，并结合我们的常规研究成果对极值波型进行分类。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，这种方法表明有时不可能计算闭合涡流发生的时间。直到现在，它还明白在涡度不变的情况下可能存在这种情况。然而，在一般涡度分布的情况下，是否存在闭合涡旋还有待证实。用有限差分方法进行了数值模拟，得到了一些结果。至于深度，它只处理了一个有限的…更多的案例。我们的研究结果如下。在下文中，我们考虑了涡度函数随处为正(或负)且随深度衰减的情况。当表面张力起作用时，但给出了涡度分布，所有的极值波型都是悬垂型或重叠型，即光滑边界。对于重力波，当涡度函数随水深衰减时，就形成了具有角点或尖点的极值波型。另一方面，在重力波的涡度不变的情况下，在到达极值波的过程中出现了一种悬垂型波，最终出现了拐角。根据我们的实验，只有在涡度不变的情况下才能看到这种悬垂类型的重力波。引力波是一种值得注意的现象，未来需要对其进行更多的研究。我们在台北中央研究院数学研究所举行的“台湾-日本非线性分析与应用数学联席会议”上公布了上述结果。它计划在整理了几条数字数据后，作为论文收集。较少

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