On rotational surfaces in 3 dimensional de Sitter space with Weingarten condition

In this article, we study spacelike and timelike rotational surfaces in a 3--dimensional de Sitter space $\mathbb{S}^3_1$ which are the orbit of a regular curve under the action of the orthogonal transformation of 4--dimensional Minkowski space $\mathbb{E}^4_1$ leaving a spacelike, a timelike or a degenerate plane pointwise fixed. We determine the profile curve of such Weingarten rotational surfaces parameterized by the principal curvature. Then, we classify spacelike and timelike Weingarten rotational surface in $\mathbb{S}^3_1$ with the principal curvatures $\kappa$ and $\lambda$ satisfying $\kappa=a\lambda+b$ or $\kappa=a\lambda^m$ for special cases of constants $a, b$ and $m$.