| Abstract: |
| The Klein-Gordon equation is a relativistic form of the Schr\"odinger equation. In the non-relativistic limit (as the speed of light goes to infinity) of Klein-Gordon equations, one derives, at least formally, Schr\"odinger equations.
We find a strong connection between the stability analysis in geometric optics and non-relativistic limit of Klein-Gordon equations. By employing the techniques in geometric optics, we obtain the optimal convergence rates. Moreover, for quadratic nonlinearities, we show the long time approximation of Klein-Gordon equations by Schr\"odinger equations in the non-relativistic limit regime.
Even in the framework of geometric optics, we find that the strong transparency conditions are not satisfied. We introduce a compatible condition and a singular localization method which allows us to prove the stability of WKB solutions over long time intervals. This compatible condition is weaker than the strong transparency condition. The singular localization method allows us to do delicate analysis near resonances. |
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