### 13.5 Concluding remarks

It is of interest to note that in the case of , the spatial variation dies out and the
solution behaves like a solution to an ODE asymptotically. On the other hand, the ODE of which it is
approximately a solution is not the ODE, which is obtained by dropping the spatial derivatives in the
original equation.
Furthermore, in the polarized case, the integrand appearing on the left-hand side of Equation (58) is
unbounded as . In fact, the best bound for the integrand is due to Equation (52). On the
other hand, the integral is conserved. Moreover, since this conserved quantity determines the overall
behavior of the solution, it is clear that the problem of analyzing the asymptotics numerically is not trivial.
The same phenomenon appears in the nonpolarized case. However, it is of interest to note that the reason
why the mathematical analysis is possible is in part due to the difference in decay rates between
and