  The weakly nonlinear regime of a viscoelastic Navier--Stokes fluid is
investigated. For the purely hydrodynamic case, it is known that large-scale
perturbations tend to the minima of a Ginzburg-Landau free-energy functional
with a double-well (fourth-order) potential. The dynamics of the relaxation
process is ruled by a one-dimensional Cahn--Hilliard equation that dictates the
hyperbolic tangent profiles of kink-antikink structures and their mutual
interactions. For the viscoelastic case, we found that the dynamics still
admits a formulation in terms of a Ginzburg--Landau free-energy functional. For
sufficiently small elasticities, the phenomenology is very similar to the
purely hydrodynamic case: the free-energy functional is still a fourth-order
potential and slightly perturbed kink-antikink structures hold. For
sufficiently large elasticities, a critical point sets in: the fourth-order
term changes sign and the next-order nonlinearity must be taken into account.
Despite the double-well structure of the potential, the one-dimensional nature
of the problem makes the dynamics sensitive to the details of the potential. We
analysed the interactions among these generalized kink-antikink structures,
demonstrating their role in a new, elastic instability. Finally, consequences
for the problem of polymer drag reduction are presented.
