Solitons and chaos: a deterministic route to the Schrödinger equation

• A new study by IFISC (UIB-CSIC) shows that the statistical behavior of particles predicted by quantum mechanics can emerge from the deterministic motion of solitons in chaotic fields.
• The work, published in Physical Review Research, provides a novel perspective on quantum tunneling and the uncertainty principle as emergent properties of nonlinear dynamics.

What is the true origin of quantum uncertainty? A new study led by Damià Gomila, researcher at the Institute for Cross-Disciplinary Physics and Complex Systems (IFISC, UIB-CSIC), proposes that the famous Schrödinger equation, the cornerstone of quantum mechanics, can be derived from the deterministic dynamics of solitons moving through a chaotic background field.

Solitons are self-sustaining waves that travel without losing their shape. First seen in a Scottish canal in the 1830s, they have since been found in water, light pulses in optical fibers, plasmas, and even theoretical models of matter. Their stability comes from a perfect balance: dispersion tries to spread the wave out while nonlinearity pulls it together. The result is a “bullet-wave” that behaves like a particle, making solitons key candidates for bridging classical and quantum behavior.

The research, published in Physical Review Research, shows that solitons in a Galilean-invariant nonlinear field theory behave like classical particles when moving in a perfect vacuum, strictly following Newton’s second law. However, when placed on a fluctuating, chaotic background, their position and momentum start to fluctuate. Remarkably, these fluctuations obey an exact uncertainty relation that naturally gives rise to the ensemble behavior described by the Schrödinger equation.

The study reveals a striking duality. On a perfectly calm background, solitons resemble rigid particles, moving deterministically under external forces. But in a chaotic environment, their ensemble dynamics no longer follow Newton alone: they reproduce the uncertainty principle and the full structure of the time-dependent Schrödinger equation. The amplitude of the background fluctuations plays the role of Planck’s constant, thereby connecting the strength of chaos to the emergence of quantum phenomena.

To test the theory, the author simulated thousands of solitons colliding against a potential barrier. While many were reflected, a fraction passed through, despite not having sufficient classical energy, a hallmark of the quantum tunneling effect. The probability of transmission measured in the simulations matched with the predictions of the Schrödinger equation.

“The results show that key features of quantum phenomena, such as uncertainty and tunneling, can be reproduced from fully deterministic soliton dynamics. Quantum mechanics may emerge as an ensemble description of solitonic particles moving in chaotic fields”, explains Damià Gomila, researcher at IFISC (UIB-CSIC) and author of the study.

The work contributes to the ongoing debate on the foundations of quantum mechanics. By grounding quantum rules on deterministic, local dynamics, the study offers an alternative to the standard Copenhagen interpretation. While focused on one-dimensional systems, the research sets the stage for extensions to higher dimensions, where solitons acquire richer properties including spin and topological structure. Future work should explore whether other fundamental aspects of quantum behavior, such as wavefunction interference, can also be explained within this deterministic soliton framework.

Gomila, Damià. “Solitons, Chaos, and Quantum Phenomena: A Deterministic Approach to the Schrödinger Equation.” Physical Review Research, vol. 7, no. 3, 2025, p. 033276. https://doi.org/10.1103/l3xp-yrrv