  We employ the Hamiltonian formalism of macroscopic fluctuation theory to
study large deviations of integrated current in the Kipnis-Marchioro-Presutti
(KMP) model of stochastic hear flow when starting from a step-like initial
condition. The KMP model belongs to the hyperbolic universality class where
diffusion remains relevant no matter how large the fluctuating current is. The
extreme current statistics for the KMP model turns out to be sub-Gaussian, as
distinguished from the super-Gaussian statistics found for the Symmetric Simple
Exclusion Process and other models of the elliptic class. The most probable
time history of the system, which dominates the extreme current statistics of
the KMP model, involves two large-amplitude solitary pulses: of the energy
density field and of the conjugate "momentum" field. The coupled pulses
propagate with a constant speed, but their amplitudes slowly grow with time, as
the energy density pulse collects most of the available energy on its way.
