  The Korteweg-de Vries and Benjamin-Ono nonlinear wave equations can describe
solitary waves, all of which propagate in the same direction and which emerge
from collisions with their shapes unchanged. There are technical challenges to
giving a description of the thermal equilibrium of a dilute gas of such
solitons. These challenges can be overcome by considering a stochastic source
of new solitons located at $x=0$. The total intensity and the momentum
distributions of the solitons emitted by such sources determine the equilibrium
distribution of soliton momenta for $x>0$.
