Numerical approach for bifurcation of nonlinear problem
Project/Area Number 
12640142

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Japan Women's University (2001) Nihon University (2000) 
Principal Investigator 
SHOJI Mayumi Japan Women's University, Department of Mathematical and Physical Sciences, Professor, 理学部, 教授 (10216161)

Project Period (FY) 
2000 – 2001

Project Status 
Completed (Fiscal Year 2001)

Budget Amount *help 
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2001: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)

Keywords  bifurcation / water wave / interfacial wave / interfacial wave / rotational wave 
Research Abstract 
The purpose of this study is to confirm whether nonsymmetric solutions exist or not on the bifurcation problem of the surface water waves and, if exist, to see their bifurcation structures. The existence of nonsymmetric solutions has not yet been proved mathematically. J. A. Zufiria ('87, '88) gave nonsymmetric solutions numerically, which are mode (1, 2, 3) waves in both cases of infinite and finite depth of fluid. However their nonsymmetricities 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 nonsymmetric 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 nonsymmetric 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 nonsymmetric 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.

Report
(3 results)
Research Products
(8 results)