  When subjected to sufficiently strong velocity gradients, solutions of long,
flexible polymers exhibit flow instabilities and chaotic motion, often referred
to as elastic turbulence. Its mechanism differs from the familiar,
inertia-driven turbulence in Newtonian fluids, and is poorly understood. Here,
we demonstrate that the dynamics of purely elastic pressure-driven channel
flows of dilute polymer solutions are organised by exact coherent structures
that take the form of two-dimensional travelling waves. Our results demonstrate
that no linear instability is required to sustain such travelling wave
solutions, and that their origin is purely elastic in nature. We show that the
associated stress profiles are characterised by thin, filament-like
arrangements of polymer stretch, which is sustained by a solitary pair of
vortices. We discuss the implications of the travelling wave solutions for the
transition to elastic turbulence in straight channels, and propose ways for
their detection in experiments.
