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

本研究的目的是研究受非线性影响的波函数的渐近行为，并描述这种渐近行为。已知如果非线性影响过大，则波函数会在有限时间内发散。另一方面，如果非线性的影响较弱，则波函数在时间上全局存在，并且随着时间趋于无穷，它趋向于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，即我们感兴趣的是摄动波函数在时间上全局存在，但随着时间趋于无穷，它不趋向于任何自由波函数的情况。为了考虑这种非线性扰动，我们的第一个任务是找到具有全局时间解的非线性波动方程，其渐近行为可能不同于相应齐次波动方程的任何解。下一步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种非线性扰动的例子。在一些例子中，波函数的渐近性优于自由解的渐近性。另一方面，它比其他例子的自由解更差。这种差异是由非线性的阶数和系数计算得出的量决定的。在前一种情况下，渐近轮廓由自由解的第二次迭代给出。另一方面，在后一种情况下，自由解的渐近轮廓与辐射场密切相关。我们得到了一个合适的常微分方程，其解给出了自由辐射场的修正。综上所述，长程型的非线性摄动是复杂的，包含了大量的变化来产生不同类型的渐近行为。少

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