On complex singularities of solutions of the equation Hux-u+up=0

It has been proved recently by Bona and Li Yi (Bona J L and Li Yi A 1997 J. Math. Pure Appl. 76 377) that solitary wave solutions of a certain class of nonlinear nonlocal equations can be extended into the complex domain. In this paper we present an approach for the study of singularities of such extensions. This approach is applied to the localized solutions of the equation Hu(x) - u + u(p) = 0, p = 3, 4, 5 where H is the Hilbert transform. The location of the closest to real axis singularity z = z(0) was found numerically; the analysis of this type of singularity shows that in the vicinity of z = z(0) these complex extensions of solutions cannot be represented by the series 1/(z-z0)(rho) Sigma (infinity)(n=0) A(n)(z-z(0))(n), i.e. they do not correspond to some power of a meromorphic function.