N-break states in a chain of nonlinear oscillators

In the present work we explore a prestretched 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 two- through five-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.