$N$-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.

Figure 1

The top panel shows stable (blue, solid) and unstable (blue, dashed) modes with one break as a function of the prestretching parameter $d$. Also shown (cyan) is the uniform stretch mode (no breaks). The inset represents examples of the profiles with an elastic (cyan), an unstable break (blue), and a stable break (red). As $d$ grows, the unstable mode merges with the uniform stretch mode. The bottom panel shows the largest two eigenvalues of the one break (again, solid for the stable part, dashed for the unstable), together with the largest eigenvalue of the elastic (uniform stretch) mode. The numbers in subscripts indicate the order of the eigenvalue, and the subscript letters indicate stability, with $s$ standing for stable and $u$ for unstable.

Figure 2

The top panel shows the amplitude of the two-break solutions as a function of prestretching, $d$. The inset represents an example of the two profiles for a given $d$. The bottom panel shows the two largest eigenvalues of this branch (for both its energy and strain increasing and decreasing segments), which is always unstable. Note that it exists only for a higher prestretching than the one-break modes. Subscripts in legend as in Fig. 1.

Figure 3

The top panel shows the amplitude of the three-break solutions as a function of prestretching, $d$. The inset represents an example of the solution profiles for a given $d$. The bottom panel shows the three largest eigenvalues associated with the saddle and center portions of the three-break branch, which are both always unstable. Subscripts in legend as in Fig. 1, but now the 3 lowest eigenvalue pairs are shown.

Figure 4

Combined results for elastic and one- to five-break branches. The top panel summarizes our results for break length, and the bottom panel shows the energy of these branches as a function of the potential prestretching parameter $d$.

Figure 5

For the one-break branch, the upper (top row) and lower (bottom row) segment eigenvectors (${\widehat{\mathbf{e}}}_{1,2}$) are shown in the left panel (the two principal ones). The right panels show the one-break solutions with $0.2\times {\widehat{\mathbf{e}}}_{1,2}.$ Here $d=1.05$, as in the inset of Fig. 1.

Figure 6

Similar to the previous figure, but now for the two-break branch, for $d=1.07$, as in Fig. 2. In the right panel, a perturbation involving the relevant eigenvectors, $0.2\times {\widehat{\mathbf{e}}}_{1,2}$, has been added to the two segments (increasing or decreasing in the top or bottom, respectively) of the branch.

Figure 7

Similar to the previous figures, but now for the three-break branch, and $d=1.07$, corresponding to the inset of Fig. 3. Here, too, a perturbation in the direction of the leading eigenvectors of the form $0.2\times {\widehat{\mathbf{e}}}_{1,2,3}$ was added to the upper (top row) and lower (bottom row) segments of the branch (middle and right columns). The left column shows the mode profile and the first three eigenvectors; the middle column shows the linear combination of the mode with the first or second eigenvector; the right column shows the linear combination of the mode with the third eigenvector.

Figure 8

The dynamical evolution of the one-break branch is shown in spatiotemporal ($n-t$) contour-plot form of the displacements. The initial condition consists of the stationary solution with a perturbation of $\pm 0.01\times {\widehat{\mathbf{e}}}_1$ added to it, for $d=1.05$. (a) Upper (linearly stable) segment branch mode $u_{1u}+0.01\times {\widehat{\mathbf{e}}}_1$; (b) upper segment branch mode $u_{1u}-0.01\times {\widehat{\mathbf{e}}}_1$; (c) lower (unstable) segment branch mode $u_{1l}+0.001\times {\widehat{\mathbf{e}}}_1$; (d) lower segment branch mode $u_{1l}-0.001\times {\widehat{\mathbf{e}}}_1$. In the last two, the instability leads, respectively, to oscillations around the upper segment branch and to degeneration to the homogeneous state.

Figure 9

Similar to Fig. 8. Here the initial condition consists of the two-break waveforms with a perturbation added in the form of the second eigenmode, $\pm 0.01\times {\widehat{\mathbf{e}}}_2$, for $d=1.07$. (a) Upper segment branch mode $u_{2u}+0.01\times {\widehat{\mathbf{e}}}_2$; (b) upper segment branch mode $u_{2u}-0.01\times {\widehat{\mathbf{e}}}_2$; (c) lower segment branch mode $u_{2l}+0.01\times {\widehat{\mathbf{e}}}_2$; (d) lower segment branch mode $u_{2l}-0.01\times {\widehat{\mathbf{e}}}_2$. Notice that although we perturb the wave in the direction of the less unstable eigenmode ${\widehat{\mathbf{e}}}_2$, the more unstable one (${\widehat{\mathbf{e}}}_1$) eventually crucially contributes to the destabilization dynamics of both segments of the two-break branch.

Figure 10

Similar to the previous figures, but now for a three-break branch with a perturbation $\pm 0.05\times {\widehat{\mathbf{e}}}_3$, for $d=1.07$. (a) Upper segment of the branch mode $u_{3u}+0.05\times {\widehat{\mathbf{e}}}_3$; (b) upper segment of the branch mode $u_{3u}-0.05\times {\widehat{\mathbf{e}}}_3$; (c) lower segment of the branch mode $u_{3l}+0.05\times {\widehat{\mathbf{e}}}_3$; (d) lower segment of the branch mode $u_{3l}-0.05\times {\widehat{\mathbf{e}}}_3$. In all four cases, eventually the dynamics results in a one-break state.

Figure 11

Same as Fig. 10, but with perturbation $\pm 0.05\times {\widehat{\mathbf{e}}}_1$, for $d=1.07$. (a) Upper segment of the branch mode $u_{3u}+0.05\times {\widehat{\mathbf{e}}}_1$; (b) upper segment of the branch mode $u_{3u}-0.05\times {\widehat{\mathbf{e}}}_1$; (c) lower segment of the branch mode $u_{3l}+0.05\times {\widehat{\mathbf{e}}}_1$; (d) lower segment of the branch mode $u_{3l}-0.05\times {\widehat{\mathbf{e}}}_1$. The resulting dynamics is more diverse, potentially leading to a homogeneous state in panel (c), the survival of a central break in panels (b) and (d), as well as the survival of one of the lateral breaks in panel (a).

Figure 12

The three-break waveform is perturbed by $\pm 0.05\times {\widehat{\mathbf{e}}}_2$, for $d=1.07$. (a) Upper segment of the branch mode $u_{3u}+0.05\times {\widehat{\mathbf{e}}}_2$; (b) upper segment of the branch mode $u_{3u}-0.05\times {\widehat{\mathbf{e}}}_2$; (c) lower segment of the branch mode $u_{3l}+0.05\times {\widehat{\mathbf{e}}}_2$; (d) lower segment of the branch mode $u_{3l}-0.05\times {\widehat{\mathbf{e}}}_2$. In all cases, one of the lateral breaks asymptotically persists.