  In this paper, a rarefaction wave under space-periodic perturbation for the 3
times 3 rate-type viscoelastic system is considered. It is shown that if the
initial perturbation around the rarefaction wave is suitably small, then the
solution of the rate-type viscoelastic system tends to the rarefaction wave.
The stability of solutions under periodic perturbations is an interesting and
important problem since the perturbation keeps oscillating at the far fields.
That is, the perturbation is not integral in space. The key of proof is to
construct a suitable ansatz carrying the same oscillation as the solution. Then
we can find cancellations between solutions and ansatz such that the
perturbation belongs to some Sobolev space. The nonlinear stability can be
obtained by the weighted energy method.
