  We investigate the decay for $|x|\rightarrow \infty$ of weak Sobolev type
solutions of semilinear nonlocal equations $Pu=F(u)$. We consider the case when
$P=p(D)$ is an elliptic Fourier multiplier with polyhomogeneous symbol $p(\xi)$
and derive sharp algebraic decay estimates in terms of weighted Sobolev norms.
In particular, we state a precise relation between the singularity of the
symbol at the origin and the rate of decay of the corresponding solutions. Our
basic example is the celebrated Benjamin-Ono equation {equation}
\label{BO}(|D|+c)u=u^2, \qquad c>0,{equation} for internal solitary waves of
deep stratified fluids. Their profile presents algebraic decay, in strong
contrast with the exponential decay for KdV shallow water waves.
