Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts

The position of an invasion front, propagating into an unstable state, fluctuates because of the shot noise coming from the discreteness of reacting particles and stochastic character of the reactions and diffusion. A recent macroscopic theory yields the probability of observing, during a long time, an unusually slow front. The theory is formulated as an effective Hamiltonian mechanics which operates with the density field and the conjugate "momentum" field. Further, the theory assumes that the most probable density field history of an unusually slow front represents, up to small corrections, a traveling front solution of the Hamilton equations. Here we verify this assumption by solving the Hamilton equations numerically for models belonging to the directed percolation universality class.