  We improve the decay argument by [Bona and Li, J. Math. Pures Appl., 1997]
for solitary waves of general dispersive equations and illustrate it in the
proof for the exponential decay of solitary waves to steady Degasperis-Procesi
equation in the nonlocal formulation. In addition, we give a method which
confirms the symmetry of solitary waves of the maximum height. Finally, we
discover how the symmetric structure is connected to the steady structure of
solutions to the Degasperis-Procesi equation, and give a more intuitive proof
for symmetric solutions to be traveling waves. The improved argument and new
methods above can be used for the decay rate of solitary waves to many other
dispersive equations and will give new perspectives on symmetric solutions for
general evolution equations.
