Solitons, originally defined as a special class of exact solutions in classical integrable systems, are now broadly employed across many areas of physics to describe spatially localized excited states that arise from the interplay between nonlinearity and dispersion, including in non-integrable systems. In condensed matter physics, representative examples include soliton lattices in chiral magnets, spinor Bose–Einstein condensates, as well as self-consistent soliton dynamics and FFLO phases in superconductors, superfluids, and organic conductors. Another major research domain is fluid mechanics, where solitons in shallow water waves and rogue waves have been extensively investigated. Related phenomena have also been investigated in nonlinear optical systems, with research additionally focusing on integrable turbulence.

From the unified perspective of established soliton theories—such as Sato theory and the algebro-geometric formalism—solitons of diverse forms observed across different physical systems are understood to reflect distinct manifestations of a common underlying mathematical structure. Consequently, cross-disciplinary research that integrates insights from multiple fields can offer novel perspectives and may even provide concrete physical realizations of some of the most profound mathematical results in soliton theory.

Recently, researchers have investigated a classical integrable equation that emerged as a byproduct of studies on superconductivity, and have identified two distinctive soliton solutions, termed “tsunami solitons” and “Korteweg–de Vries (KdV) rocks.” These solutions have been proposed as potential models for hydrodynamic phenomena in disordered backgrounds. Starting from a theoretical framework for parity-mixed superconductors, a hierarchy of soliton equations was derived using Krichever’s method. By solving these equations in the presence of a superconducting gap, the authors constructed tsunami solitons, which exhibit propagating step-like structures accompanied by turning-back dynamics and oscillatory behavior, as well as immobile KdV rocks.

These integrable equations are found to be closely related to models describing other physical systems, including plasmas and multilayer fluids. Moreover, the unexpectedly large space of stationary solutions—allowing arbitrary configurations of multiple KdV rocks—gives rise to soliton dynamics in disordered backgrounds. The origin of these phenomena can be understood within the framework of algebraic geometry. The concept of “isodispersive phases” has also been introduced.

This line of research opens a new interdisciplinary direction spanning condensed matter physics, plasma physics, fluid dynamics, and soliton theory, and indicates promising avenues for future applications in disordered physical systems. Shall we embark on this exciting cross-disciplinary journey with solitons?

(Written by Daisuke A. Takahashi on behalf of all authors)

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