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A Hamiltonian approach to the CH-KP equation

This paper derives the CH-KP equation, a mildly nonlinear model for quasi-unidirectional shallow water waves, from Euler equations using Hamiltonian reduction and long-wave asymptotics, while also analyzing its Hamiltonian structure and providing examples of weak peakon-type solutions.

Original authors: R. Camassa, G. Falqui, E. Sforza

Published 2026-08-04

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CC BY 4.0

Original authors: R. Camassa, G. Falqui, E. Sforza

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

Imagine the ocean not as a flat, calm mirror, but as a living, breathing entity where waves dance, crash, and sometimes even break in surprising ways. For centuries, scientists have tried to write the "rules of the dance" for these water waves using complex math. One of the most famous rulebooks is called the Kadomtsev-Petviashvili (KP) equation. Think of it as a master instruction manual that explains how long, gentle waves move across a wide pond, especially when they wiggle a little bit side-to-side. This manual is so powerful that it can be simplified to explain many other famous wave behaviors, like the classic solitary waves (solitons) that travel without changing shape.

However, real water isn't always perfectly smooth. Sometimes, waves get so steep that they form sharp, jagged peaks instead of rounded hills. In the world of one-dimensional waves (waves moving in a straight line), there is a special rulebook called the Camassa-Holm (CH) equation. This book is famous for describing these "peakons"—waves that look like sharp mountain peaks rather than smooth curves. These peakons are special because they can bounce off each other like elastic billiard balls, a behavior that standard wave rules can't quite capture. The big question scientists have been asking is: What happens if we take these sharp, peaky waves and let them move in two dimensions, wiggling side-to-side just like the KP equation allows? Does the "sharpness" survive the extra dimension, or does the math break down?

This paper, titled "A Hamiltonian approach to the CH-KP equation," sets out to answer that question by building a new, more complete rulebook. The authors, a team of mathematicians from the US and Italy, start with the most fundamental laws of fluid motion (the Euler equations) and use a clever mathematical trick called "Hamiltonian reduction" to strip away the unnecessary complexity. They are essentially taking a 3D model of two layers of fluid (like oil floating on water) and shrinking it down to a 2D surface, while carefully keeping track of the energy and momentum that make waves behave the way they do.

The result of their hard work is a new equation they call the CH-KP equation. This equation is a hybrid: it combines the side-to-side wiggle of the classic KP equation with the sharp, peaky nature of the CH equation. The authors show that this new equation is not just a random guess; it has a deep, elegant mathematical structure (a "Hamiltonian structure") that proves it is a valid and consistent description of nature. They didn't just write the equation down; they derived it step-by-step from the ground up, ensuring that the physics of energy and momentum are perfectly preserved.

But the paper doesn't stop at just writing the equation. The authors also play with it to see what kind of waves it creates. They found that, just like in the one-dimensional world, this new equation supports "peakons"—waves with sharp peaks. They even discovered that these peaky waves can travel in two dimensions, and their crests can be tilted at an angle, like a wave moving diagonally across a pool. The team simulated what happens when two of these 2D peakons collide. In some cases, they crash into each other and exchange energy, creating a complex, shifting pattern. In other cases, they pass through each other, leaving a "phase shift" (a change in position) behind, much like the billiard ball analogy.

The authors are careful to note that while these solutions are mathematically beautiful and consistent with their new equation, they are currently exploring them as "weak solutions." This is a technical term meaning they are valid mathematical answers, but they involve sharp corners that require special handling in the math. The paper suggests that these peakons could be used as building blocks to understand more complex wave patterns, similar to how Lego bricks build a castle. However, the authors admit that fully understanding how these waves behave in the real world, and whether they are stable over long periods, is a job for future research. For now, they have successfully built the bridge between the sharp world of peaky waves and the wide world of two-dimensional water motion, providing a new, robust tool for scientists to study the ocean's most energetic moments.

Technical Summary: A Hamiltonian Approach to the CH-KP Equation

Problem Statement
The paper addresses the derivation and analysis of the CH-KP equation, a two-dimensional extension of the Camassa-Holm (CH) equation that incorporates weak transverse dependence, analogous to how the Kadomtsev-Petviashvili (KP) equation extends the Korteweg-de Vries (KdV) equation. While the CH equation is well-known in one dimension for its higher-order mixing of nonlinearity and dispersion, its capacity to support peakon solutions and wave breaking, and its bi-Hamiltonian structure, the two-dimensional extensions have a comparatively short history. Previous derivations of the CH-KP equation relied on classical expansions of 3D water wave equations. This work seeks to provide an alternative, rigorous derivation based on the Hamiltonian picture of water wave theory, specifically utilizing Dirac reduction techniques on stratified fluid systems. Additionally, the paper aims to characterize families of solutions, particularly "peakon" solutions (weak solutions with peaked crests), and analyze their collision dynamics in two dimensions.

Methodology
The authors employ a systematic asymptotic and Hamiltonian reduction strategy:

Hamiltonian Framework for Stratified Fluids: The derivation begins with the Euler equations for a two-layer, sharply stratified incompressible fluid. Utilizing the Bowman-Benjamin (BB) Hamiltonian representation for heterogeneous flows, the system is reduced to a set of canonical variables: the interface height $\zeta$ and the horizontal momentum shear $\mu$ .
Air-Water Limit and SGN Equations: The authors perform a limit where the density of the upper fluid vanishes and its height becomes unbounded. This recovers the Serre-Green-Naghdi (SGN) equations (also known as Su-Gardner or Green-Naghdi equations) in a Hamiltonian formalism. The energy functional is expanded in terms of long-wave asymptotics ( $\epsilon = h/L \ll 1$ ).
Weak Transversality and Unidirectionalization: A weak transverse scaling is introduced ( $y = \beta y'$ with $\beta \ll \epsilon$ ). The momentum variable is transformed from the interface value to a velocity $u^*$ evaluated at a specific height dependent on the wave amplitude. This transformation is crucial for obtaining a unidirectional model.
Dirac Reduction: The core of the derivation involves a Dirac reduction process. By imposing a constraint that relates the surface elevation $\zeta$ to the velocity $u^*$ (effectively selecting a specific wave direction), the authors reduce the constrained Hamiltonian system. This yields a reduced Poisson tensor and a corresponding evolution equation for the variable $m = u^* - u^*_{xx}$ .
Scaling and Canonical Form: Through specific asymptotic equivalences and variable scalings, the derived equation is transformed into the canonical CH-KP form. The resulting equation is identified as the KP-II type, featuring a specific balance between nonlinearity and dispersion.
Solution Analysis: The paper investigates solutions by taking the limit where a smoothing parameter vanishes ( $\hat{k} \to 0$ ), reducing the system to a form admitting peakon solutions. The authors generalize the $N$ -peakon ansatz from the 1D CH equation to the 2D CH-KP case, deriving the finite-dimensional Hamiltonian system governing the evolution of peakon positions and amplitudes.

Key Contributions and Results

Derivation of CH-KP: The paper successfully derives the CH-KP equation from the Euler equations for stratified fluids using a Hamiltonian reduction approach. This provides a geometric and structural justification for the equation, distinct from previous expansion-based derivations.
Hamiltonian Structure: The authors explicitly construct the bi-Hamiltonian structure of the CH-KP equation. They identify two conserved quantities, $E$ and $F$ , and the associated Hamiltonian operators $P_1$ and $P_2$ , confirming the equation's integrability properties in the 1D limit and its structural similarity to the CH equation.
Peakon Solutions: The study demonstrates that in the limit $\hat{k} \to 0$ , the CH-KP equation admits peakon solutions. These are weak solutions of the form $U(x, y, t) = \sum p_i(t) e^{-|x - b y - q_i(t)|}$ .
Collision Dynamics: The paper analyzes the interaction of peakons in two dimensions.
- Tilted Crests: For traveling wave solutions of the form $U(x, y, t) = U(x - by - ct)$, the peakon crests are tilted by an angle $\arctan(b)$ in the $(x,y)$ -plane, where $b$ is a slope parameter.
- Collision Scenarios: Detailed analysis of $N=2$ peakon collisions reveals complex behaviors. For instance, a peakon-antipeakon collision results in the annihilation of the wave at the collision point ( $t=0$ ) followed by re-emergence and exchange of positions.
- Transverse Dependence: The authors explore how the transverse coordinate $y$ acts as a parameter in the evolution of peakon amplitudes and positions. They present examples where the initial shift $d_0$ depends on $y$ (e.g., linear or quadratic dependence), leading to "cross-shaped" crest interactions or asymmetric collision geometries that differ significantly from the 1D case.

Significance and Claims
The paper claims that the CH-KP equation represents a "mildly nonlinear version" of the KP equation, offering a different balance between nonlinearity and dispersion compared to the standard KP-II equation. The primary significance lies in the rigorous Hamiltonian derivation, which connects the CH-KP equation directly to the fundamental Euler equations for stratified fluids via the SGN intermediate model.

The authors emphasize that while the 1D peakon theory is well-established as a generalized basis for constructing smooth solutions and a particle-based numerical scheme, the role of peakons in the 2D CH-KP context is less clear. They provide examples of 2D peakon interactions to illustrate the richness of the solution space, noting that while these solutions are mathematically interesting, their utility as a generalized basis for arbitrary 2D initial data remains an open question. The paper concludes by noting that stability issues and extensions to fully stratified fluids with nonlocal operators are important future directions, but are beyond the scope of the current work. The derivation confirms that the CH-KP equation belongs to the KP-II class, distinguishing it from the KP-I case where different results have been recently obtained.

Technical Summary: A Hamiltonian Approach to the CH-KP Equation

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