Numerical approach for bifurcation of nonlinear problem

非线性问题分岔的数值方法

基本信息

批准号：
12640142
负责人：
SHOJI Mayumi
金额：
$ 0.96万
依托单位：
Japan Women's University (2001)Nihon University (2000)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640142/
关键词：
bifurcation water wave interfacial wave interfacial wave rotational wave

项目摘要

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.

这项研究的目的是确认是否存在非对称溶液在地表水波的分叉问题上，并且（如果存在）查看其分叉结构。非对称解决方案的存在尚未在数学上证明。 J. A. Zufiria（'87，88）在数值上给出了非对称溶液，在液体的无限和有限深度中，它是模式（1、2、3）波的模式（1、2、3）。但是，它们的非对称性是如此微小，它们的分叉结构晦涩难懂。因此，我们想通过自己的算法调查他的结果。我们执行了以下方案：1。我们继续通过修改程序来计算，该程序已将其用于无旋波或旋转波的分叉问题2。如果我们在上述计算中失败，我们会尝试采用另一种方法。对1。我们尚未获得任何非对称解决方案，但事先是结论。我们需要更严格和深刻的模拟，因为这是一个非常微妙的问题。今年，我们主要研究2种方法。这是研究表面波问题的概括的界面渐进波问题。在面部间波的情况下，证明存在模式的三分叉点（L，M，N）。可以将Zufiria的非对称模式（1、3、6）的非对称波解释为面部间波的三分叉的效果，因为表面波问题嵌入了界面问题中。我们编程代码来计算界面波问题并模拟一些分叉结构。我们尚未通过这种方法获得任何非对称解决方案。但是，随着关键参数的变化，面部波的分叉结构的一些变化是我们的结果。研究围绕三重分叉的结构将很有趣，这仍然是我们的目标。