  The spatiotemporal propagation behavior of a solitary wave is investigated on
a Fermi-Pasta-Ulam ring. We observe the emergence of a cnoidal wave excited by
the solitary wave. The cnoidal wave may coexist with the solitary wave for a
long time associated with the periodic exchange of energy between these two
nonlinear waves. The module of the cnoidal wave, which is considered as an
indicator of the nonlinearity, is found to oscillate with the same period of
the energy exchange. After the stage of coexistence, the interaction between
these two nonlinear waves leads to the destruction of the cnoidal wave by the
radiation of phonons. Finally, the interaction of the solitary wave with
phonons leads to the loss of stability of the solitary wave.
