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Nonreciprocal Breathing Solitons

Jonas Veenstra1,*, Oleksandr Gamayun2,*, Martin Brandenbourger1,3,*, Freek van Gorp1, Hans Terwisscha-Dekker1, Jean-Sébastien Caux1, and Corentin Coulais1

  • *These authors contributed equally to this work.
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Phys. Rev. X 15, 031045 – Published 18 August, 2025

DOI: https://doi.org/10.1103/nrv2-9h8z

Abstract

Breathing solitons consist of a fast beating wave within a compact envelope of stable shape and velocity. They can propagate and carry information and energy in a variety of contexts such as plasmas, optical fibers, and cold atoms, but propagating breathers have remained elusive when energy conservation is broken. Here, we report on the observation of breathing, unidirectional, arbitrarily long-lived solitons in nonreciprocal, nonconservative active metamaterials. Combining precision desktop experiments, numerical simulations, and perturbation theory on generalizations of the sine-Gordon and nonlinear Schrödinger equations, we demonstrate that unidirectional breathers generically emerge in weakly nonlinear nonreciprocal materials, and that their dynamics are governed by an unstable fixed point. Crucially, breathing solitons can persist for arbitrarily long times provided that (i) this fixed point displays a bifurcation when a delicate balance between energy injection and dissipation is struck and (ii) the initial conditions allow the dynamics to reach this bifurcation point. Importantly, discrete effects tend to stabilize these nonreciprocal breathers over a wider range of initial conditions. Our work establishes nonreciprocity as a promising avenue to generate stable nonlinear unidirectional waves and could be generalized beyond metamaterials to optics, soft matter, and superconducting circuits.

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Popular Summary

Unlike typical waves that spread out and fade, solitons travel like particles, maintaining their wavelike shape over long distances. However, in real-world systems—where friction or energy input is unavoidable—ideal solitons usually do not survive. In this study, we tackle the challenge of stabilizing solitons under such nonideal conditions and find a way to create and sustain a special kind of soliton known as a “breathing” soliton, even in systems where energy is not conserved.

We build a system made of active mechanical oscillators and explore how nonreciprocal interactions—where one part of the system responds differently than another—affect soliton behavior. Unlike in symmetrical systems, here agent A influences agent B differently than B influences A. This asymmetry turns out to be key. It allows the soliton to accelerate and then settle into a steady, unchanging motion, all without losing its shape or energy. This behavior had been particularly hard to achieve with breathing solitons, which constantly change their form as they move.

Our results show that combining nonreciprocal and nonlinear dynamics enables the existence of robust, long-lived solitons in energy-dissipating systems. These waves could be used to carry signals or energy efficiently, making them promising for applications in sensors, energy-harvesting devices, and robotic systems. Looking ahead, this work opens new possibilities for engineering smart materials that rely on stable, self-sustaining wave motion—a step toward adaptive systems that operate reliably in dynamic environments.

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Phys. Rev. X 15, 031045– Published 18 August, 2025
  • Received 17 May 2024
  • Accepted 10 July 2025
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