Asymptotics of solutions of the Klein-Gordon equation

The Klein–Gordon-equation is a relativistic wave equation, related to the Schrödinger equation. In one space dimension and, it is possible to choose units (in which the speed of light is LaTeX: 1 ) so that the equation is

$LaTeX: \frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + u = 0$ .

In an MSc thesis from $LaTeX: V\left(x\right)u$ , asymptotics as $LaTeX: \sqrt{t^2-x^2}\to \infty$ for this and related equations were studied, by using a method developed by Lars Hörmander which relies on the method of stationary phase. In addition, similar asymptotics for the equation with an added potential term $LaTeX: V\left(x\right)u$ were derived.

In this project, we will build on this work, and derive such asymptotics where the fixed potential $LaTeX: V\left(x\right)$ is replaced by a moving potential $LaTeX: V\left(x-ct\right)$ , with LaTeX: |c|<1 .

Kontakt: Sara Maad Sasane