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Using branch continuation in the FENE-P model, we show that finite-amplitudetravelling waves borne out of the recently-discovered linear instability ofviscoelastic 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 weakindicating 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 asaddle 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-viscosityratio $\beta=0.9$. In this latter creeping flow limit, we also show that thesewaves exist at $Wi \lesssim 50$ for higher polymer concentrations - $\beta \in[0.5,0.97)$ -- where there is no known linear instability. Our resultstherefore indicate that these travelling waves -- found in simulations andnamed `arrowheads' by Dubief et al. {\em arXiv}.2006.06770 (2020) - exist muchmore generally in $(Wi,Re, \beta)$ parameter space than their spawning neutralcurve 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.

Finite-amplitude elastic waves in viscoelastic channel flow from large to zero Reynolds number