Numerical approach for bifurcation of nonlinear problem

点击翻译按钮获取中文摘要

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.