  Prominent examples of longitudinal phase separation in elastic systems
include elastic necking, the propagation of a bulge in a cylindrical party
balloon and the beading of a gel fiber subject to surface tension. Here we
demonstrate that, if the parameters of such a system are tuned near a critical
point (where the difference between the two phases vanishes) then the behaviour
of all systems is given by the minimization of a simple and universal elastic
energy familiar from Ginzburg-Landau theory in an external field. We minimize
this energy analytically, which yields not only the well known interfacial tanh
solution, but also the complete set of stable and unstable solutions in both
finite and infinite length systems, unveiling the elastic system's full shape
evolution and hysteresis. Correspondingly, we also find analytic results for
the the delay of onset, changes in criticality and ultimate suppression of
instability with diminishing system length, demonstrating that our simple
near-critical theory captures much of the complexity and choreography of
far-from-critical systems. Finally, we find critical points for the three
prominent examples of phase separation given above, and demonstrate how each
system then follows the universal set of solutions.
