🌀 nonlinear sciences

Dissipation-driven champion solitons in one-dimensional shallow-water waves

This paper demonstrates that in a bidirectional shallow-water wave field governed by the Kaup-Boussinesq equation, weak high-wavenumber dissipation combined with random waves drives the formation of "champion solitons" through successive interactions, revealing a counterintuitive mechanism where dissipation intensifies rather than damps coherent structures.

Original authors: Ashleigh Simonis, Sergey Nazarenko, Jalal Shatah, Yulin Pan

Published 2026-08-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ashleigh Simonis, Sergey Nazarenko, Jalal Shatah, Yulin Pan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet, rhythmic motion of water, there exists a hidden order that defies the usual chaos of the ocean. For over a century, scientists have studied solitary waves—single, self-contained humps of water that travel long distances without losing their shape. These are not the crashing breakers of a storm, but rather smooth, persistent travelers that emerge when the force pushing the water forward balances perfectly with the tendency of the wave to spread out. In the idealized world of mathematics, these waves behave like perfect billiard balls: they can collide with one another, pass right through, and emerge unchanged, retaining their original speed and height. This predictable behavior is known as integrability, a property where the system conserves its energy and structure indefinitely. However, the real world is rarely ideal. In nature, water is subject to friction, turbulence, and random gusts of wind that introduce dissipation, or energy loss. Conventional wisdom suggests that when you add this kind of damping to a system, it should simply wear everything down, smoothing out the waves until they vanish. The question that has long puzzled researchers is whether this gentle erosion could ever do something unexpected, perhaps even strengthening a wave instead of destroying it.

A team of researchers has now simulated a specific scenario in shallow water to answer this question, revealing a surprising mechanism where dissipation actually helps create a "champion" wave. Using a mathematical model that describes how water moves in both directions, the scientists introduced a weak form of damping that targets only the smallest, fastest ripples in the water. They began with a chaotic mix of random waves and a scattering of solitary waves moving in opposite directions. In a world without this damping, the solitary waves would simply bounce off each other and continue on their paths, never changing their fundamental nature. But with the weak damping in place, the story changed dramatically. The system did not simply fade away. Instead, it went through a distinct sequence of events where the random waves and smaller solitary waves slowly lost energy, while one specific wave in each direction began to grow stronger.

This process unfolded in three clear stages. First, the entire system relaxed as the damping took hold, causing the number of waves to drop and their heights to decrease. Then, a counterintuitive phase of intensification began. As the smaller waves continued to lose energy, one solitary wave in the left-moving group and one in the right-moving group started to absorb energy from their neighbors. These two waves, which the researchers call "champion solitons," grew larger and faster, leaving the others behind. They did not grow by magic or by pulling energy from the air; they grew by interacting with the other waves in a specific way. The simulations showed that when a faster champion wave caught up to a slower, smaller wave moving in the same direction, they would travel together for a long time. During this prolonged partnership, the larger wave effectively siphoned energy from the smaller one, growing stronger while the smaller one faded. This transfer of energy was the key to their survival and growth.

Crucially, the researchers found that this phenomenon would not happen if the water contained only the solitary waves. When they ran the simulation with just the solitary waves and the damping, no champion emerged; the waves simply faded away without any dramatic growth. The presence of the random, chaotic background waves was essential. These random waves acted as a catalyst, feeding the small-scale ripples that the damping could act upon, which in turn allowed the energy transfer between the solitary waves to occur. Without this chaotic background, the system remained too quiet for the champions to form. The result was a single, dominant wave in each direction that grew to a height roughly three times the average size of the waves in the system, a scale that would classify it as a rogue wave in the ocean.

After reaching their peak, these champion waves entered a quasi-steady phase where they traveled for a long time with nearly constant speed and height, maintaining a delicate balance between the energy they gained from smaller waves and the energy they lost to the damping. Eventually, even these powerful waves began to decay as the damping effects took their toll, but they had lived much longer and grown much larger than they would have in a purely random or purely damped environment. The study suggests that in shallow water, where random waves, coherent structures, and friction coexist, the very force we expect to dampen extreme events might actually be the engine that drives them. By targeting the smallest scales of the water's motion, weak dissipation can trigger a chain reaction that concentrates energy into a single, massive wave, offering a new explanation for how rogue waves might form in nature.

Technical Summary: Dissipation-driven champion solitons in one-dimensional shallow-water waves

Problem Statement
The paper investigates the long-time evolution of a bidirectional shallow-water wave field modeled by the Kaup-Boussinesq (KB) equation. While the integrable KB system is known for elastic soliton interactions and isospectrality (where solitons preserve shape and speed after collisions), the authors introduce a weak, high-wavenumber dissipative perturbation to break this integrability. The central problem is to determine how such dissipation, typically expected to damp coherent structures, affects the statistical distribution of solitons and whether it can induce the formation of a single dominant "champion" soliton in each propagation direction.

Methodology
The study employs numerical simulations of the dimensionless KB system on a periodic domain ( $L = 1024\pi$ ) using a pseudospectral method with a fourth-order Runge-Kutta time-marching scheme (IF-RK4).

Initial Conditions: The system is initialized with a Gaussian random wave spectrum containing both left- and right-propagating components.
Perturbation: A weak damping term is added to the Fourier space equations, targeting high wavenumbers ( $k > k_p$ ) via a filter function $\Lambda(k)$ . This mimics physical small-scale damping without modifying the nonlinear terms of the governing equations.
Analysis Tool: The Direct Scattering Transform (DST) is utilized at each time step to decompose the wave field into discrete eigenvalues (representing solitons/bound states) and continuous spectrum (representing radiation/random waves). Solitons are identified by discrete eigenvalues $\zeta$ satisfying $|c_s| > 1$ , where $c_s$ is the soliton velocity.
Comparative Experiments: To isolate mechanisms, the authors compare: (1) the integrable KB system (no dissipation), (2) a dissipative system with a random wave background, and (3) a dissipative system containing only an ensemble of solitons (no random waves).

Key Contributions and Results
The authors identify a three-stage evolution process leading to the formation of "champion solitons" (solitons that grow to dominate the field):

Relaxation Stage: Initially, both the random wave background and the soliton population decay due to dissipation. The number of bound states decreases, and their amplitudes reduce, but the ratio of soliton energy to total energy remains roughly constant.
Intensification Stage: Following the initial decay, a counter-intuitive growth phase occurs. While the total number of solitons continues to decrease, the peak amplitude of specific solitons increases significantly. This stage is driven by inelastic collisions where energy is transferred from smaller, co-propagating solitons to larger ones. The process culminates in the emergence of a single "champion" soliton in each direction of propagation. These champions reach amplitudes approximately three times the significant wave height ( $\eta_{max}/H_s \approx 3$ ), classifying them as rogue waves.
Quasi-Steady Stage: The champion solitons propagate with nearly constant amplitude for a prolonged period ( $O(10^5 T_p)$ ), maintaining a balance between energy loss from dissipation and energy gain from interactions with subdominant solitons. Eventually, they decay as the reservoir of smaller solitons is depleted.

Mechanistic Insights

Role of Random Waves: Crucially, the paper demonstrates that dissipation alone is insufficient to generate champion solitons. In simulations with only an ensemble of solitons (no random background), collisions remain nearly elastic, and no champion forms. The random wave component is essential as it:
- Provides the high-wavenumber spectral content necessary for the weak dissipation to act effectively.
- Acts as an energy reservoir, sustaining smaller solitons and facilitating the energy transfer mechanisms required for intensification.
Interaction Dynamics: Energy transfer is most efficient during co-propagating interactions (overtaking collisions) rather than counter-propagating ones. Co-propagating solitons with similar velocities remain coupled for extended periods, allowing the dominant soliton to draw energy from the subdominant one. Counter-propagating collisions are too brief ( $O(1 T_p)$ ) to facilitate significant energy exchange.

Significance
The paper claims that this work identifies a new mechanism for rogue wave formation in shallow-water settings. The findings challenge the conventional intuition that dissipation universally suppresses coherent structures; instead, weak high-wavenumber dissipation, when coupled with a random wave background, can induce the intensification of solitons. This mechanism is distinct from dissipative solitons (which require a balance of gain and loss) and differs from previous studies on champion solitons that relied on modifying nonlinear terms. The authors suggest that because high-wavenumber dissipation is a common feature in physical systems, this mechanism may be relevant to the generation of extreme ocean events in natural environments.

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