2007 Fiscal Year Final Research Report Summary
On the asymptotic behavior of solution to systems of nonlinear wave equations of long range type
| Project/Area Number |
17540157
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Osaka University |
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Principal Investigator |
KUBO Hideo Osaka University, Graduate Sinai of Science, Associated professor (50283346)
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| Co-Investigator(Kenkyū-buntansha) |
HAYASHI Nakao Osaka University, Graduate School of Sciencie, Professor (30173016)
MATSUMURA Akitaka Osaka University, Graduate School of Information Science and Technology, Professor (60115938)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Analisys / Pratial differential equations / Nonlinear wave |
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| Research Abstract |
The aim of this research is to study the asymptotic behavior of wave functions perturbed by the influence of the nonlinearity and characterize such asymptotic behavior. It is known that if the influence of the nonlinearity is too strong, then the wave function diverges in a finite time. On the other hand, if the influence of the nonlinearity is weak, then the wave function exists globally in time and it tends to a wave function which is free from the nonlinear perturbation in the sense of the energy as time goes to infinity.
In this research, we treat the intermediate case, namely, we are interested in the case where the perturbed wave function exists globally in time, but it does not tend to any free wave function as time goes to infinity. In order to consider such nonlinear perturbation, our first task is to find nonlinear wave equations which admit global in time solutions whose asymptotic behavior may differ from any solution to the corresponding homogeneous wave equations. Then the
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next step is to show that its asymptotic behavior is actually different from the free solution. As for these problems, we seemed to find several examples of such nonlinear perturbation. For some examples, the asymptotic behavior of the wave function is better compared with that of the free solution. On the other hand, it is worse than that of the free solution for the other examples. Such difference is determined by a quantity which is computed from the order and the coefficients of the nonlinearity.
In the former case, the asymptotic profile is given by a second iterate of the free solution. On the other hand, in the latter case, the asymptotic profile is closely related to the radiation field for the free solution. We obtain a suitable ordinary differential equation whose solution gives the modification of the free radiation field.
In conclusion, the nonlinear perturbation of long range type is complicated and contains a full of variety to produce different kinds of asymptotic behavior. Less
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