We show that the problem of a directed polymer on a tree with disorder can be
reduced to the study of nonlinear equations of reaction-diffusion type. These
equations admit traveling wave solutions that move at all possible speeds above
a certain minimal speed. The speed of the wavefront is the free energy of the
polymer problem and the minimal speed corresponds to a phase transition to a
glassy phase similar to the spin-glass phase. Several properties of the polymer
problem can be extracted from the correspondence with the traveling wave:
probability distribution of the free energy, overlaps, etc.