  The recently-discovered centre-mode instability of rectilinear viscoelastic
shear flow (Garg et al. Phy. Rev. Lett. 121, 024502, 2018) has offered an
explanation for the origin of elasto-inertial turbulence (EIT) which occurs at
lower Weissenberg ($Wi$) numbers. In support of this, we show using weakly
nonlinear analysis that the subcriticality found in Page et al. (Phys. Rev.
Lett. 125, 154501, 2020) is generic across the neutral curve with the
instability only becoming supercritical at low Reynolds ($Re$) numbers and high
$Wi$. We demonstrate that the instability can be viewed as purely elastic in
origin even for $Re=O(10^3)$, rather than `elasto-inertial', as the underlying
shear does not energise the instability. It is also found that the introduction
of a realistic maximum polymer extension length, $L_{max}$, in the FENE-P model
moves the neutral curve closer to the inertialess $Re=0$ limit at a fixed ratio
of solvent-to-solution viscosities, $\beta$. In the dilute limit ($\beta
\rightarrow 1$) with $L_{max} =O(100)$, the linear instability can brought down
to more physically-relevant $Wi\gtrsim 110$ at $\beta=0.98$, compared with the
threshold $Wi=O(10^3)$ at $\beta=0.994$ reported recently by Khalid et al.
(arXiv: 2103.06794) for an Oldroyd-B fluid. Again the instability is
subcritical implying that inertialess rectilinear viscoelastic shear flow is
nonlinearly unstable - i.e. unstable to finite amplitude disturbances - for
even lower $Wi$.
