Numerical approach for bifurcation of nonlinear problem

这项研究的目的是确定表面水波分叉问题是否存在非对称解，如果存在，则查看其分叉结构。非对称解的存在性尚未得到数学证明。J.A.Zufiria(‘87，’88)在数值上给出了非对称解，在无限深和有限深两种情况下都是(1，2，3)波。然而，它们的非对称性很小，分叉结构也很模糊。因此，我们希望通过我们自己的算法来研究他的结果。我们执行了以下方案：1.通过修改我们用于无旋波或旋波分叉问题的程序来继续计算。如果我们在上面的计算中失败了，我们将尝试另一种方法。关于1.我们还没有得到任何非对称解，但它是预先得出结论的。由于这是一个非常微妙的问题，我们需要更严格和更深入的模拟。今年，我们主要研究2的另一种方法，即研究界面逐行波问题，它是表面波问题的推广。在界面波的情况下，证明了(L，m，n)模存在三个分岔点。可以将Zufiria的(1，3，6)模的非对称波解释为界面波的三重分叉的影响，因为表面波问题嵌入到界面问题中。我们编写了计算界面波问题的程序，并模拟了一些分叉结构，但还没有得到任何非对称解。然而，我们的结果是，随着关键参数的变化，界面波的分叉结构也发生了一些变化。研究三重分叉周围的结构将是很有趣的，它仍然是我们的目标。

The purpose of this study is to confirm whether non-symmetric solutions exist or not on the bifurcation problem of the surface water waves and, if exist, to see their bifurcation structures. The existence of non-symmetric solutions has not yet been proved mathematically. J. A. Zufiria ('87, '88) gave non-symmetric solutions numerically, which are mode (1, 2, 3) waves in both cases of infinite and finite depth of fluid. However their non-symmetricities are so minute and their bifurcation structures are obscure. So we would like to investigate his results by our own algorithms.We carried out the following schemes :1. We continue to compute by modifying our programs, which we have used for the bifurcation problem of irrotational waves or rotational waves.2. If we fail in the above computation, we try to do another approach.Regarding 1. we have not yet obtained any non-symmetric solutions, but it is beforehand to conclude. We need much more strict and profound simulations since it is a very delicate problem.This year, we study mainly another approach of 2. It is to study the interfacial progressive wave problem that is a generalization of the surface wave problem. In the case of inter facial waves, it is proved that there exist triple bifurcation points of mode (l, m, n). It might be possible to interpret Zufiria's non-symmetric waves of mode (1, 3, 6) as the effect of the triple bifurcation of inter facial waves, because the surface wave problem is embedded in the interfacial problem. We programd codes to compute the interfacial wave problem and simulated some bifurcation structures.We have not yet obtained any non-symmetric solution by this approach. However it is our results to see some changes of bifurcation structure of inter facial waves as the key parameter varies. It would be interesting to study structures around the triple bifurcation and it is still our target.