  For a simple one dimensional lattice version of a travelling wave equation,
we obtain an exact relation between the initial condition and the position of
the front at any later time. This exact relation takes the form of an inverse
problem: given the times $t_n$ at which the travelling wave reaches the
positions $n$, one can deduce the initial profile. We show, by means of complex
analysis, that a number of known properties of travelling wave equations in the
Fisher-KPP class can be recovered, in particular Bramson's shifts of the
positions. We also recover and generalize Ebert-van Saarloos' corrections
depending on the initial condition.
