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On the asymptotic behavior of solution to systems of nonlinear wave equations of long range type

长程型非线性波动方程组解的渐近行为

基本信息

批准号：
17540157
负责人：
KUBO Hideo
金额：
$ 2.3万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

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 … More 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

本研究的目的是研究受非线性影响的波函数的渐近行为，并刻画这种渐近行为。众所周知，如果非线性的影响太大，那么波函数就会在有限时间内发散。另一方面，如果非线性的影响很弱，那么波函数在时间上是全局存在的，并且随着时间的推移，它趋于一个不受能量意义上的非线性扰动的波函数。在本研究中，我们处理中间情况，即我们感兴趣的情况，即扰动波函数在时间上全局存在，但随着时间的推移不趋于任何自由波函数。为了考虑这种非线性扰动，我们的第一个任务是找到在时间上允许整体解的非线性波动方程，其渐近行为可能不同于相应的齐次波动方程的任何解。然后是…下一步就是证明它的渐近行为实际上不同于自由解。对于这些问题，我们似乎找到了几个这样的非线性扰动的例子。对于某些例子，波函数的渐近行为比自由解的渐近行为更好。另一方面，对于其他例子，它比自由解的解要差。这种差异是由一个量决定的，这个量是由非线性的阶数和系数计算出来的。在前一种情况下，渐近曲线是由自由解的第二次迭代给出的。另一方面，在后一种情况下，自由解的渐近分布与辐射场密切相关。我们得到了一个合适的常微分方程解，它的解给出了自由辐射场的修正。总之，长程型的非线性摄动是复杂的，包含了各种各样的渐近行为。较少