Solitonic waves – waves that keep their shape and direction of motion for a long time – have intrigued physicists for almost two centuries. In real-world circumstances, these waves eventually die out due to energy loss. A team of UvA physicists have now discovered how a particular type of interaction can be used to create very stable solitons, even in circumstances where energy is not conserved.
An unusual type of wave

In 1834, John Scott Russell observed an unusual phenomenon in the Union Canal in Scotland. After a moving boat had come to a halt, the water wave that the boat had caused continued moving through the canal, keeping virtually the same speed and the same shape. It took more than half a century, until the work of Dutch mathematicians Diederik Korteweg and Gustav de Vries in 1895, before the phenomenon that Russell observed had been explained in all its mathematical detail. What Russell had seen was a ‘solitary wave’, a phenomenon now better known as a soliton.

Today, we know that such solitons do not only occur in shallow water, but also in optics, in magnetic fields, and in many other branches of physics where wave phenomena play a role. Unlike typical waves that spread out and fade, solitons travel like particles, maintaining their wave-like shape over long distances. However, in real-world systems—where friction or energy input is unavoidable—ideal solitons usually don’t survive. If John Russell had kept following the wave caused by the boat, eventually it would have faded away.

Asymmetric interaction

In their latest research, published in the journal Physical Review X this week, a team of physicists at the University of Amsterdam tackled the challenge of stabilizing solitons under such nonideal, real-world conditions. Walking in the footsteps of their UvA-predecessors Korteweg and De Vries, first author Jonas Veenstra and his six collaborators have now found a way to create a special kind of soliton known as a “breathing” soliton, and sustain it over a long period of time, even in systems where energy is not conserved.

To achieve this, in the lab of group leader Corentin Coulais, the researchers built a system made of ‘active mechanical oscillators’ – small rods that can rotate and are powered by individual tiny motors, connected through rubber bands. When one oscillator starts moving, so does its neighbour, and in this way waves can form – including solitonic ones. The particular oscillators that were used in the experiments have an interesting property: they can influence each other in a nonreciprocal way. That is, an oscillator influences its neighbour differently than the neighbour influences the original oscillator. When the physicists explored how these nonreciprocal interactions affected soliton behavior, they found that this asymmetry was the key to long-lived solitons. The asymmetry allowed the solitonic wave to accelerate and then settle into a steady, unchanging motion, all without losing its shape or energy.

From observation to application

The observed behavior had always been particularly hard to achieve with so-called ‘breathing’ solitons, which constantly change their form as they move. This time, this was precisely what the researchers saw: a sustained, breathing soliton. While not completely new – the group already observed the first breather soliton experimentally 6 years ago – it took a new and more precise experimental setup developed since then to pin down the exact conditions under which breather solitons travel in a stable manner under non-reciprocity.

The results show the existence of robust, long-lived solitons, even in systems where energy gets lost. This is not just a pretty phenomenon that one can observe in the lab, similar to how John Russell observed the very first soliton in a canal two centuries ago. Sustained solitonic waves are very useful: they could be used to carry signals or energy efficiently, making them promising for applications in sensors, energy-harvesting devices, and robotic systems.

With such real-world applications in mind, Veenstra and colleagues are currently trying to go beyond mere chains of oscillators, investigating how similar waves behave in two-dimensional surfaces of non-reciprocal oscillators. Looking ahead, their research opens new possibilities for engineering smart materials that rely on stable, self-sustaining wave motion – a step toward adaptive systems that operate reliably in all kinds of dynamic environments.

Publication

Nonreciprocal Breathing Solitons, Jonas Veenstra, Oleksandr Gamayun, Martin Brandenbourger, Freek van Gorp, Hans Terwisscha-Dekker, Jean-Sébastien Caux, and Corentin Coulais, Phys. Rev. X 15, 031045.