  Using a matrix product state algorithm with infinite boundary conditions, we
compute high-resolution dynamic spin and quadrupolar structure factors in the
thermodynamic limit to explore the low-energy excitations of isotropic
bilinear-biquadratic spin-1 chains. Haldane mapped the spin-1 Heisenberg
antiferromagnet to a continuum field theory, the non-linear sigma model
(NL$\sigma$M). We find that the NL$\sigma$M fails to capture the influence of
the biquadratic term and provides only an unsatisfactory description of the
Haldane phase physics. But several features in the Haldane phase can be
explained by non-interacting multi-magnon states. The physics at the
Uimin-Lai-Sutherland point is characterized by multi-soliton continua. Moving
into the extended critical phase, we find that these excitation continua
contract, which we explain using a field-theoretic description. New excitations
emerge at higher energies and, in the vicinity of the purely biquadratic point,
they show simple cosine dispersions. Using block fidelities, we identify them
as elementary one-particle excitations and relate them to the integrable
Temperley-Lieb chain.
