N-break states in a chain of nonlinear oscillators

TL;DR

This paper investigates the existence and stability of multi-break solutions in a pre-stretched nonlinear oscillator chain with Lennard-Jones interactions, revealing stability transitions and eigenvalue structures.

Contribution

It provides the first analysis of stability and bifurcation structures of multi-break solutions in nonlinear oscillator chains with Lennard-Jones potentials.

Findings

One break solutions change stability at a critical precompression strain.

Higher break solutions have increasing unstable eigenvalue pairs.

Numerical simulations confirm the stability and instability predictions.

Abstract

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.