13.5 Concluding remarks

It is of interest to note that in the case of A2 + 4BC < 0, 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 (58View Equation) is unbounded as t → ∞. In fact, the best bound for the integrand is Ct1 ∕2 due to Equation (52View Equation). 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 ⟨Pt⟩2 and

∫ P2d 𝜃. S1 t


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