Holomorphic extension of solutions of semilinear elliptic equations

We prove, for a wide class of semilinear elliptic differential and pseudodifferential equations in ${\mathbb{R}}^d$, that the solutions which are sufficiently regular and have a certain decay at infinity extend to holomorphic functions in sectors of ${\mathbb{C}}^d$, improving earlier results where the extension was shown for a strip. Moreover, exponential decay for such extended solutions is also proved. The results apply, in particular, to solitary wave solutions of many classical nonlinear evolution equations.