Numerical solution of some unstable problem

While modelling nonlinear wakefield excited with a short laser pulse in a plasma channel, the pulse group speed is normally less than the speed of light, whereas they are usually considered to be equal. This paper shows that consideration of the speed difference leads to an exponential unstability of the model. It is also proved that this unstable problem is equivalent to a specific stable problem. This feature allows us to develop stable numerical schemes for their solution.