  Superconducting cosmic strings can exhibit longitudinal, pinching
instabilities in some regions of the parameter space. We make predictions about
the onset of this instability using the thin string approximation (TSA) and
develop an improved analysis that remains applicable for small wavelength
perturbations, where the TSA breaks down. We use simulations of perturbed
strings to assess the accuracy of the TSA, test the predictions of our new
analysis and demonstrate an improvement over previous methods in the
literature. Notably, it appears that the instabilities are typically present
for a larger range of magnetic strings than previously expected, and we show
examples of pinching instabilities also occurring in electric strings. However,
both our simulations and predictions agree that strings near the chiral limit
are free from pinching instabilities and in particular our results support our
previously published claim that vortons can be stable to all classical
perturbations if they are sufficiently large.
