  {\bf Exact} solutions of the string equations of motion and constraints are
{\bf systematically} constructed in de Sitter spacetime using the dressing
method of soliton theory. The string dynamics in de Sitter spacetime is
integrable due to the associated linear system. We start from an exact string
solution $q_{(0)}$ and the associated solution of the linear system $\Psi^{(0)}
(\lambda)$, and we construct a new solution $\Psi(\lambda)$ differing from
$\Psi^{(0)}(\lambda)$ by a rational matrix in $\lambda$ with at least four
poles $\lambda_{0},1/\lambda_{0},\lambda_{0}^*,1/\lambda_{0}^*$. The periodi-
city condition for closed strings restrict $\lambda _{0}$ to discrete values
expressed in terms of Pythagorean numbers. Here we explicitly construct solu-
tions depending on $(2+1)$-spacetime coordinates, two arbitrary complex numbers
(the 'polarization vector') and two integers $(n,m)$ which determine the string
windings in the space. The solutions are depicted in the hyperboloid coor-
dinates $q$ and in comoving coordinates with the cosmic time $T$. Despite of
the fact that we have a single world sheet, our solutions describe {\bf multi-
ple}(here five) different and independent strings; the world sheet time $\tau$
turns to be a multivalued function of $T$.(This has no analogue in flat space-
time).One string is stable (its proper size tends to a constant for $T\to\infty
$, and its comoving size contracts); the other strings are unstable (their
proper sizes blow up for $T\to\infty$, while their comoving sizes tend to cons-
tants). These solutions (even the stable strings) do not oscillate in time. The
interpretation of these solutions and their dynamics in terms of the sinh-
Gordon model is particularly enlighting.
