  Using branch continuation in the FENE-P model, we show that finite-amplitude
travelling waves borne out of the recently-discovered linear instability of
viscoelastic channel flow (Khalid et al. {\em J. Fluid Mech.} {\bf 915}, A43,
2021) are substantially subcritical reaching much lower Weissenberg ($Wi$)
numbers than on the neutral curve at a given Reynolds ($Re$) number over $Re
\in [0,3000]$. The travelling waves on the lower branch are surprisingly weak
indicating that viscolastic channel flow is susceptible to (nonlinear)
instability triggered by small finite amplitude disturbances for $Wi$ and $Re$
well below the neutral curve. The critical $Wi$ for these waves to appear in a
saddle node bifurcation decreases monotonically from, for example, $\approx 37$
at $Re=3000$ down to $\approx 7.5$ at $Re=0$ at the solvent-to-total-viscosity
ratio $\beta=0.9$. In this latter creeping flow limit, we also show that these
waves exist at $Wi \lesssim 50$ for higher polymer concentrations - $\beta \in
[0.5,0.97)$ -- where there is no known linear instability. Our results
therefore indicate that these travelling waves -- found in simulations and
named `arrowheads' by Dubief et al. {\em arXiv}.2006.06770 (2020) - exist much
more generally in $(Wi,Re, \beta)$ parameter space than their spawning neutral
curve and hence can either directly, or indirectly through their instabilities,
influence the dynamics seen far away from where the flow is linearly unstable.
Possible connections to elastic and elasto-inertial turbulence are discussed.
