Soliton-like solutions of the Camassa--Holm equation with variable coefficients and a small dispersion

This paper constructs and proves the asymptotic accuracy of one-phase and two-phase soliton- and peakon-like solutions for the Camassa–Holm equation with variable coefficients and small dispersion by expressing them as sums of a regular background and a singular component, supported by explicit examples.

The Gist

Imagine the ocean. Usually, when we think of waves, we imagine smooth, rolling hills of water that travel across the sea. In the world of mathematics, these are called solitons. They are special because they are like perfect, indestructible surfers: they can crash into each other, bounce off, and keep going without losing their shape or speed.

But there's another type of wave, a bit more rebellious, called a peakon. Instead of a smooth hill, a peakon looks like a sharp mountain peak or a jagged saw-tooth. It's a wave with a pointy top. The famous Camassa-Holm (CH) equation is the mathematical rulebook that describes how these waves behave.

The Problem: The Ocean is Changing

In the real world, the ocean isn't uniform. The depth changes, the currents shift, and the wind varies from place to place and time to time. The original CH equation assumes a perfectly flat, unchanging ocean. But what if we want to model waves in a river that gets wider, or a sea where the temperature changes?

The authors of this paper, Yuliia and Valerii Samoilenko, asked: "What happens to these perfect waves (solitons and peakons) if the environment they travel through is constantly changing?"

They looked at a modified version of the equation with variable coefficients (changing rules) and small dispersion (a tiny bit of "fuzziness" or spreading).

The Solution: Building Waves with LEGO

Since the math for these changing environments is incredibly messy and impossible to solve with a single, neat formula, the authors used a clever strategy called asymptotic expansion.

Think of it like building a complex sculpture out of LEGO bricks:

1. The Background (The Regular Part): First, they build the "base" or the background scenery. This represents the general state of the water (like a calm sea or a slow current) that exists everywhere. This part is smooth and predictable.
2. The Wave (The Singular Part): On top of that background, they add the special "wave" bricks. This is the soliton or the peakon. This part is "singular," meaning it's the intense, concentrated energy of the wave.

The magic of their method is that they can build this sculpture layer by layer. They start with the main shape, then add a tiny correction layer, then another even tinier one, and so on. The more layers they add, the more accurate their model becomes.

The Two Types of Waves They Studied

1. The Smooth Surfer (Soliton-like)

• The Shape: A smooth, rounded wave.
• The Challenge: When the environment changes, the math gets tricky. The authors found that for a single wave (one-phase), they could describe the main shape perfectly, even if it was hidden inside a complex formula (like a secret code). They proved that they could keep adding correction layers to make the prediction as accurate as they wanted.
• The Two-Wave Dance (Two-Phase): When two of these waves interact, they usually pass through each other and keep going. The authors managed to describe the main shape of this two-wave interaction, but the math got so complicated that they couldn't easily add the tiny correction layers. It's like they could draw the two surfers high-fiving, but couldn't quite calculate the exact ripples on the water afterward.

2. The Pointy Mountain (Peakon-like)

• The Shape: A sharp, pointy wave.
• The Challenge: These are harder because the wave has a "kink" or a sharp corner. In math, sharp corners are annoying because they break the usual rules of smoothness.
• The Trick: The authors treated the wave as two separate pieces: the left side of the peak and the right side. They solved the math for each side individually and then "glued" them together at the sharp point.
• The Result: Surprisingly, for these pointy waves, they could actually find the exact shape of the main wave and all the tiny correction layers, even in a changing environment. It's like they found a way to perfectly predict the shape of a jagged lightning bolt even if the wind was blowing it around.

Why Does This Matter?

You might ask, "Who cares about math equations for waves?"

• Real-World Physics: These equations help us understand real phenomena like tsunamis, tidal bores, and even how certain fluids (like paint or blood) flow.
• Predicting the Future: By understanding how these waves behave in changing conditions, scientists can build better models to predict natural disasters or design better ships.
• Mathematical Beauty: It shows that even when the rules of the universe get messy and changeable, there are still underlying patterns and structures that we can uncover with the right tools.

The Takeaway

The authors successfully created a "recipe" for predicting how special waves behave in a chaotic, changing world. They showed that even when the environment is unpredictable, we can still build accurate models by breaking the problem down into a smooth background and a concentrated wave, layer by layer. They proved that while some wave interactions are too complex to solve perfectly, others (especially the pointy peakons) can be understood with incredible precision.

In short: They figured out how to track the surfers and the jagged peaks even when the ocean floor is shifting beneath them.

Technical Summary

1. Problem Statement

The paper investigates the Camassa–Holm (CH) equation with variable coefficients (vcCH) and a small dispersion parameter. The classical CH equation is a well-known integrable system describing shallow water waves, famous for admitting two distinct types of solitary wave solutions: smooth solitons and peaked solitons known as peakons.

The authors address the generalized equation:
$$a(x, t, \varepsilon)u_t + b(x, t, \varepsilon)uu_x - \varepsilon^2 (u_{txx} + 2u_x u_{xx} + u u_{xxx}) = 0$$
where $a$ and $b$ are variable coefficients expanded in powers of a small parameter $\varepsilon$ (representing small dispersion), and $\varepsilon \to 0$.

The Core Challenge:
Unlike the constant-coefficient CH equation, the presence of variable coefficients generally prevents the derivation of exact analytical solutions. The goal is to construct asymptotic solutions that retain the qualitative features of solitons and peakons (localized waves, specific interaction properties) in the limit of small dispersion. A significant difficulty arises because the variable coefficients break the gauge equivalence found in the Korteweg-de Vries (KdV) equation, making the construction of multi-phase solutions significantly more complex.

2. Methodology

The authors employ the nonlinear WKB (Wentzel-Kramers-Brillouin) method, adapted for singularly perturbed partial differential equations with variable coefficients.

Key Methodological Steps:

1. Asymptotic Expansion: The solution $u(x, t, \varepsilon)$ is decomposed into a regular part (background wave) and a singular part (capturing the soliton/peakon features):
   $$u(x, t, \varepsilon) = \sum_{j=0}^N \varepsilon^j [u_j(x, t) + V_j(x, t, \tau)] + O(\varepsilon^{N+1})$$
   where $\tau = (x - \phi(t))/\varepsilon$ is the stretched phase variable, and $\phi(t)$ is the phase function.

2. Functional Spaces: To handle the singular nature of the solutions, the authors define specific functional spaces:

   • $G_0, G_1$: Spaces for soliton-like solutions where terms decay rapidly at infinity or approach a constant background.
   • $G_\pm$: Spaces for peakon-like solutions, defined by "gluing" functions from the left ($\tau < 0$) and right ($\tau > 0$) of the peak, ensuring continuity but allowing derivative discontinuities.

3. Separation of Terms:

   • Regular Terms ($u_j$): Determined by solving quasi-linear and linear first-order PDEs using the method of characteristics. These represent the smooth background.
   • Singular Terms ($V_j$): Determined by solving third-order PDEs restricted to the phase curve $\Gamma$ (where $x = \phi(t)$).
     • For solitons, the main singular term is found in implicit form via a system of ODEs.
     • For peakons, the main singular term is found in explicit form (exponential decay).

4. Solvability Conditions: The existence of higher-order singular terms is established using orthogonality conditions derived from the theory of pseudodifferential operators. This ensures that the constructed asymptotic series satisfies the original equation to a specific order of accuracy.

3. Key Contributions and Results

A. One-Phase Soliton-like Solutions

• Main Term: The leading singular term $V_0$ is derived in implicit form. It satisfies a nonlinear ODE system that cannot be solved explicitly for general variable coefficients.
• Phase Function: The phase function $\phi(t)$ is constrained by algebraic inequalities involving the coefficients $a_0, b_0$ and the background $u_0$.
• Higher Order Terms: The authors prove that higher-order singular corrections ($V_1, \dots, V_N$) can be constructed explicitly in suitable functional spaces, provided orthogonality conditions are met.
• Accuracy: Theorem 2.2 and 2.3 establish that the constructed $N$-th approximation satisfies the vcCH equation with accuracy $O(\varepsilon^N)$.

B. Two-Phase Soliton-like Solutions

• Complexity: Constructing multi-phase solutions for vcCH is significantly harder than for the variable-coefficient KdV (vcKdV) equation due to the lack of gauge equivalence.
• Restriction: The authors successfully construct the main singular term ($V_0$) only under the specific assumption that the coefficients $a_0$ and $b_0$ are constant along the phase curves.
• Result: The main term is expressed implicitly using the known two-soliton solution of the standard CH equation, transformed via new variables. Higher-order terms remain an open technical challenge due to the complexity of the resulting third-order PDEs.

C. One-Phase Peakon-like Solutions

• Explicit Construction: Unlike solitons, the main singular term for peakons is found explicitly as $V_0 \sim e^{-\alpha|\tau|}$.
• Phase Function: The phase function $\phi(t)$ is determined by a specific nonlinear ODE derived from the peakon ansatz.
• Gluing: The solution is constructed by solving separate ODEs for $\tau > 0$ and $\tau < 0$ and "gluing" them at $\tau=0$ to ensure continuity.
• Accuracy: Theorem 4.1 proves the existence of asymptotic peakon-like solutions with arbitrary accuracy $O(\varepsilon^N)$.

D. Two-Phase Peakon-like Solutions

• Explicit Solution: Leveraging the simpler structure of peakons, the authors derive an explicit exact form for the main two-phase singular term.
• Method: By reducing the governing equation to the standard CH equation under specific coefficient constraints, they utilize the known two-peakon solution of the classical CH equation.
• Result: The solution describes two interacting peaks that separate and propagate independently, with the main term given in closed form.

4. Significance and Conclusion

• Generalization: The paper successfully extends the theory of soliton and peakon solutions from the constant-coefficient CH equation to the more physically realistic variable-coefficient case.
• Methodological Advancement: It demonstrates that while the nonlinear WKB method is powerful, its application to the CH equation with variable coefficients requires specific adaptations, particularly regarding the functional spaces and the handling of implicit vs. explicit singular terms.
• Limitations Identified: The study highlights a fundamental difference between the CH and KdV equations. The KdV equation's "gauge equivalence" allows for easier construction of multi-phase solutions with variable coefficients. The CH equation lacks this property, making the construction of multi-phase solitons (unlike peakons) extremely difficult and requiring restrictive assumptions on the coefficients.
• Validation: The theoretical results are validated through non-trivial examples where explicit approximate solutions are derived, and their behavior is illustrated via graphs, confirming the stability and accuracy of the asymptotic expansions.

In summary, this work provides a rigorous mathematical framework for analyzing wave propagation in non-uniform media governed by the Camassa–Holm equation, offering explicit and implicit asymptotic formulas for both soliton and peakon dynamics in the small dispersion limit.