  We consider semilinear equations of the form p(D)u=F(u), with a locally
bounded nonlinearity F(u), and a linear part p(D) given by a Fourier
multiplier. The multiplier p(\xi) is the sum of positively homogeneous terms,
with at least one of them non smooth. This general class of equations includes
most physical models for traveling waves in hydrodynamics, the Benjamin-Ono
equation being a basic example. We prove sharp pointwise decay estimates for
the solutions to such equations, depending on the degree of the non smooth
terms in p(\xi). When the nonlinearity is smooth we prove similar estimates for
the derivatives of the solution, as well holomorphic extension to a strip, for
analytic nonlinearity.
