Numerical approach for bifurcation of nonlinear problem

本研究的目的是确定表面水波分支问题是否存在非对称解，如果存在，则观察其分支结构。非对称解的存在性尚未得到数学证明。J. A. Zufiria（'87，'88）给出了非对称的数值解，在无限深和有限深两种情况下都是（1，2，3）型波。然而，它们的非线性度很小，分叉结构也不清楚。因此，我们想用我们自己的算法来研究他的结果。我们继续通过修改我们的程序来计算，我们已经使用了无旋波或有旋波的分叉问题。如果我们在上面的计算中失败了，我们尝试做另一种方法。我们还没有得到任何非对称解，但可以提前下结论。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。研究界面行波问题是表面波问题的推广。在界面波的情况下，证明了存在（l，m，n）型的三重分歧点。由于表面波问题是嵌入在界面问题中的，因此可以将Zufiria的（1，3，6）型非对称波解释为界面波的三重分叉效应。我们编制了计算界面波问题的程序，并模拟了一些分叉结构，用这种方法还没有得到任何非对称解。然而，我们的结果看到一些变化的分歧结构的界面波的关键参数的变化。这将是有趣的研究周围的三重分歧的结构，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.