  We study the self-similar motion of a string in a self-similar spacetime by
introducing the concept of a self-similar string, which is defined as the world
sheet to which a homothetic vector field is tangent. It is shown that in
Nambu-Goto theory, the equations of motion for a self-similar string reduce to
those for a particle. Moreover, under certain conditions such as the
hypersurface orthogonality of the homothetic vector field, the equations of
motion for a self-similar string simplify to the geodesic equations on a
(pseudo) Riemannian space. As a concrete example, we investigate a self-similar
Nambu-Goto string in a spatially flat Friedmann-Lema\^itre-Robertson-Walker
expanding universe with self-similarity and obtain solutions of open and closed
strings, which have various nontrivial configurations depending on the rate of
the cosmic expansion. For instance, we obtain a circular solution that evolves
linearly in the cosmic time while keeping its configuration by the balance
between the effects of the cosmic expansion and string tension. We also show
the instability for linear radial perturbation of the circular solutions.
