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

An exactly solvable travelling wave equation in the Fisher-KPP class