Travelling front of a decaying brane in string field theory

We consider the inhomogeneous decay of an unstable D-brane of bosonic string theory in a linear dilaton background in a light-cone frame. At the lowest level, the dynamical equation that describes this process is a generalisation (that includes nonlocality and time delay) of a reaction-diffusion equation studied by Fisher (and others). We argue that the equation of motion of the cubic open string field theory is satisfied at least to the second order when we start with this 'Fisher deformation', a marginal operator which has a simple pole term in its OPE. We also compute the one-point functions of closed string operators on the disc in the presence of this deformation.