jats:pUsing 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 jats:italicet al.</jats:italic>, jats:italicJ. Fluid Mech.</jats:italic>, vol. 915, 2021, A43) are substantially subcritical reaching much lower Weissenberg (jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline1.png" /> jats:tex-math$Wi$</jats:tex-math> </jats:alternatives> </jats:inline-formula>) numbers than on the neutral curve at a given Reynolds (jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline2.png" /> jats:tex-math$Re$</jats:tex-math> </jats:alternatives> </jats:inline-formula>) number over jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline3.png" /> jats:tex-math$Re\in [0,3000]$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The travelling waves on the lower branch are surprisingly weak indicating that viscoelastic channel flow is susceptible to (nonlinear) instability triggered by small finite-amplitude disturbances for jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline4.png" /> jats:tex-math$Wi$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline5.png" /> jats:tex-math$Re$</jats:tex-math> </jats:alternatives> </jats:inline-formula> well below the neutral curve. The critical jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline6.png" /> jats:tex-math$Wi$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for these waves to appear in a saddle node bifurcation decreases monotonically from, for example, jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline7.png" /> jats:tex-math$\approx 37$</jats:tex-math> </jats:alternatives> </jats:inline-formula> at jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline8.png" /> jats:tex-math$Re=3000$</jats:tex-math> </jats:alternatives> </jats:inline-formula> down to jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline9.png" /> jats:tex-math$\approx 7.5$</jats:tex-math> </jats:alternatives> </jats:inline-formula> at jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline10.png" /> jats:tex-math$Re=0$</jats:tex-math> </jats:alternatives> </jats:inline-formula> at the solvent-to-total-viscosity ratio jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline11.png" /> jats:tex-math$\beta =0.9$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this latter creeping flow limit, we also show that these waves exist at jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline12.png" /> jats:tex-math$Wi\lesssim 50$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for higher polymer concentrations, jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline13.png" /> jats:tex-math$\beta \in [0.5,0.97)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where there is no known linear instability. Our results therefore indicate that these travelling waves, found in simulations and named ‘arrowheads’ by Dubief jats:italicet al.</jats:italic> (jats:italicPhys. Rev. Fluids</jats:italic>, vol. 7, 2022, 073301), exist much more generally in jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S002211202200831X_inline14.png" /> jats:tex-math$(Wi,Re,\beta )$</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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.</jats:p>