We've largely failed to notice inherently nonlinear phenomena.
One day in 1834, Scottish naval engineer John Scott Russell was riding alongside a canal near Edinburgh, when two horses drawing a boat on the canal suddenly stopped. As the boat came to rest, it pushed up a broad swell of water, which set off down the canal, with a fascinated Russell giving chase. “I followed it on horseback,” Russell reported, “and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long... after a chase of one or two miles I lost it in the windings of the channel. Such was my first-chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.”
Russell began experimenting with such waves, and soon guessed a relationship between their speed v and height h, depending on the depth of the channel d: v = √g(h + d). Thirty-two years later, Lord Rayleigh derived this expression from basic hydrodynamics, confirming that Russell had discovered a fundamentally new and essentially nonlinear kind of wave-like excitation. Since then, our knowledge of solitons and other nonlinear waves has grown enormously, especially through the mathematics of the 'inverse scattering transform'. Yet we might still be closer to the beginning of a long exploration than to its end, with many surprises yet to come.
Take the curious 'intrinsically localized modes' (ILMs), a subject of growing interest. Numerous experiments and simulations at the macroscopic level demonstrate that a periodic array of elements with generic nonlinear restoring forces can support localized excitations, which are naturally stable and extend over a handful of lattice sites. Recent experiments with arrays of micromechanical cantilevers, for example, illustrate how an initially homogeneous excitation can grow unstable and decay into a small number of localized excitations, which persist and carry most of the vibrational energy (M. Sato et al. Rev. Mod. Phys. 78, 137–157; 2006). These localized excitations act more or less like particles, and interact strongly on close encounters, unlike traditional solitons, which pass through one another, suffering only a phase shift.
Similar ILMs have been observed in nonlinear photonic crystals and in arrays of Josephson junctions. More intriguingly, numerous studies have found indirect traces of ILMs in real-world condensed-matter systems such as charge-transfer salts, quasi-one-dimensional ferromagnets and proteins. When we think of excitations in a crystalline solid lattice, we still think in habitually linear terms; nonlinearity tends to masquerade as linear excitations plus interactions. Consequently, it seems, we've largely failed to notice inherently nonlinear phenomena.
These localized objects bring to mind the beautiful 'oscillons' discovered several years ago as stable, particle-like excitations that can form on the surface of a vibrating bed of metallic beads. Might such structures play an important role in crystalline materials even in equilibrium? Based on current understanding, Sato et al. suggest conditions might be right in some special cases — in materials exhibiting paraelectric to ferroelectric phase transitions, for example. But we might ultimately find that ILM's, like Russell's 'wave of translation', have been with us for a long time — we just haven't got used to seeing them.
Rights and permissions
About this article
Cite this article
Buchanan, M. Wave of translation. Nature Phys 2, 575 (2006). https://doi.org/10.1038/nphys395