While the round spheres are exceptionally rigid among compact self-expanding solitons, even in the class of rotational hypersurfaces, there are interesting examples of complete noncompact self-expanders. Indeed, G. Huisken and T. Ilmanen constructed a complete, rotationally symmetric, self-expander with one asymptotically cylindrical end. First, we use the shooting method to construct new self-expanders, so called infinite bottles, having two cylindrical ends (joint with G. Drugan and G. Wheeler, 2015). Second, motivated by the role of Jacobi fields for constant mean curvature surfaces, we investigate the linearized operator of the soliton equation to establish the rigidity of rotational self-expanders (joint work with G. Drugan and F. Fong, 2016).