  In the present work we explore a pre-stretched oscillator chain where the
nodes interact via a pairwise Lennard-Jones potential. In addition to a
homogeneous solution, we identify solutions with one or more (so-called)
`breaks', i.e., jumps. As a function of the canonical parameter of the system,
namely the precompression strain $d$, we find that the most fundamental one
break solution changes stability when the monotonicity of the Hamiltonian
changes with $d$. We provide a proof for this (motivated by numerical
computations) observation. This critical point separates stable and unstable
segments of the one break branch of solutions. We find similar branches for 2
through 5 break branches of solutions. Each of these higher `excited state'
solutions possesses an additional unstable pair of eigenvalues. We thus
conjecture that $k$ break solutions will possess at least $k-1$ (and at most
$k$) pairs of unstable eigenvalues. Our stability analysis is corroborated by
direct numerical computations of the evolutionary dynamics.
