A study of variational single solitary waves governed by the conservative-extended KdV equation with applications to shallow water dispersive shocks

This paper employs a variational approach based on averaged Lagrangians to derive simple, accurate single solitary wave solutions for the energy-conserving extended KdV equation and validates their effectiveness in modeling both classical and resonant dispersive shock waves in shallow water through comparison with numerical simulations.

The Gist

Imagine the ocean as a giant, chaotic dance floor. Usually, when a wave moves, it spreads out, loses energy, and breaks apart, much like a crowd dispersing after a concert. But sometimes, nature creates a special kind of wave called a solitary wave (or soliton). Think of this as a single, perfect dancer who can glide across the entire floor without losing their shape or speed, even after bumping into other dancers.

For a long time, scientists used a famous mathematical rule called the KdV equation to predict how these waves behave. It's like a reliable map for a flat, calm ocean. However, real oceans (and other fluids like liquid crystals or plasmas) are more complex. They have hidden currents and "friction" effects that the old map doesn't account for. When these extra effects are strong, the old map fails, and the waves start behaving strangely—sometimes breaking apart or shooting out energy like a lighthouse beam.

The New Map: The "Extended" KdV

The authors of this paper, Saleh Baqer and Hamid Said, created a new, more detailed map called the extended KdV (eKdV) equation. This new map includes extra terms to account for those complex, real-world effects.

However, this new map is very complicated to read. It's like trying to solve a Rubik's cube while riding a rollercoaster. Previous methods to find the shape of these special waves involved heavy algebra and complex approximations that were hard to use for practical problems.

The "Variational" Shortcut

The authors decided to try a different approach. Instead of solving the complex equations directly, they used a method called variational calculus based on "averaged Lagrangians."

The Analogy:
Imagine you want to find the fastest route for a car to drive from point A to point B, but the road has hills, valleys, and wind.

• The Old Way: You calculate the exact physics of every single molecule of air and every bump in the road. It's accurate but takes forever.
• The Authors' Way: They look at the "average" energy of the car over the whole trip. They ask, "What path minimizes the total effort?" This gives them a very good guess of the route without needing to calculate every tiny detail.

Using this "average energy" trick, they found a simple, clean formula for the shape of these solitary waves. Their solution looks like a smooth, bell-shaped hill (mathematically, a sech² profile). It's much simpler than previous attempts and easier to use for engineers and scientists who need to predict wave behavior quickly.

Testing the Map: Two Types of Shocks

To prove their new map works, they tested it on two different types of "traffic jams" in the water, known as Dispersive Shock Waves (DSWs).

1. The Classic Traffic Jam (Classical DSW):
   Imagine a sudden wave of water hitting a calm area. It forms a smooth, expanding train of waves. The authors used their simple formula to predict how fast the front of this wave train moves and how tall the leading wave is.

   • Result: Their predictions matched computer simulations almost perfectly. It's like their new map predicted the traffic jam's speed and size exactly right.

2. The Resonant Traffic Jam (Non-Classical or CDSW):
   This is the tricky part. Sometimes, the leading wave moves at just the right speed to "resonate" with the water ahead of it, like a singer hitting a note that shatters a glass. This causes the wave to shed energy (radiation) ahead of it, creating a chaotic, unstable situation.

   • The Challenge: Standard maps break down here because the wave is interacting with its own "echo."
   • The Solution: The authors combined their simple wave formula with a concept called Whitham shocks (a way to handle sudden jumps in wave properties). They treated the leading wave and the radiation ahead of it as two different zones that need to be connected.
   • Result: Even in this chaotic, resonant scenario, their simple formula predicted the behavior of the waves and the speed of the shock front with excellent accuracy.

The Bottom Line

The paper claims that by using a clever "average energy" shortcut, they found a simple, accurate way to describe complex water waves that previous methods struggled to handle.

• What they did: They derived a simple formula for solitary waves in a complex fluid model that conserves energy.
• Why it matters: This formula is much easier to use than previous complex solutions.
• Proof: They showed that when they used this simple formula to predict how waves behave in two different scenarios (normal shocks and complex, resonant shocks), the results matched high-powered computer simulations very closely.

In short, they found a "shortcut" to understanding complex wave physics that is both simple to write down and powerful enough to predict real-world behavior accurately.

Technical Summary

Technical Summary: Variational Single Solitary Waves in the Conservative-Extended KdV Equation

Problem Statement
The Korteweg–de Vries (KdV) equation is a foundational model for shallow water waves and nonlinear dispersive phenomena. However, it is often insufficient for capturing complex hydrodynamic features observed in nature and experiments, such as resonant radiation, non-monotonic wave speed dependencies, and "table-top" solitary wave profiles. These phenomena arise in regimes requiring higher-order nonlinear and dispersive terms, leading to the extended KdV (eKdV) equation.

A significant challenge in analyzing the eKdV equation is the lack of exact solitary wave solutions that remain close to the classical $\text{sech}^2$ profile while accounting for higher-order effects. Previous approaches using higher-order asymptotics, algebraic methods, or near-identity transformations (e.g., mapping to the Gardner equation) have yielded solutions that are either mathematically complex or difficult to integrate into modulation theory frameworks. Furthermore, while the eKdV equation generally admits energy conservation only under specific coefficient constraints ($c_2 = 2c_3$), a corresponding Lagrangian and Hamiltonian formulation for this specific "conservative-eKdV" case had not been explicitly established in prior literature, hindering the application of variational methods.

Methodology
The authors employ a variational approach based on the method of averaged Lagrangians to derive approximate single solitary wave solutions for the conservative-eKdV equation. The methodology proceeds as follows:

1. Model Formulation: The study focuses on the eKdV equation under the condition $c_2 = 2c_3$, which ensures the existence of an exact energy conservation law. The authors explicitly derive the associated Lagrangian density and Hamiltonian functional for this specific case, establishing a rigorous variational framework.
2. Variational Derivation: A trial solution of the form $u = a_s \text{sech}^2(w_s \theta_s)$ is substituted into the averaged Lagrangian. By applying the calculus of variations to the averaged Lagrangian with respect to the amplitude parameter $a_s$ and the inverse width $w_s$, the authors derive explicit expressions for the solitary wave velocity $V_s$ and width $w_s$ up to order $O(\epsilon)$.
3. Height Correction: To determine the actual solitonic height $A_s$ (which differs from the fixed parameter $a_s$ due to higher-order nonlinearities), the authors utilize an energy method. This involves integrating the eKdV equation and applying boundary conditions to derive a velocity-amplitude relation, which is then expanded to express $A_s$ in terms of $a_s$.
4. Application to Dispersive Shocks: The derived variational solutions are applied to two distinct classes of dispersive shock wave (DSW) problems:
   • Classical DSWs: Using the "DSW equal amplitude approximation," the authors model the shock as a train of nearly equal amplitude solitary waves.
   • Non-Classical (Resonant) DSWs (CDSW): For regimes where the leading edge emits resonant radiation (Cross-over DSWs), the authors combine the variational soliton solution with the concept of "Whitham shocks." This involves matching the bore (modeled as a solitary wavetrain) to a resonant Stokes wave ahead of the shock using modulation jump conditions derived from mass and energy conservation laws.

Key Contributions

• Variational Solitary Wave Solution: The paper presents, for the first time to the authors' knowledge, a single solitary wave solution for the conservative-eKdV equation derived via the calculus of variations. This solution is notably simpler and more analytically tractable than previous approximations obtained through algebraic or asymptotic techniques.
• Lagrangian-Hamiltonian Structure: The work explicitly constructs the Lagrangian and Hamiltonian formulations for the conservative-eKdV model ($c_2 = 2c_3$), filling a gap in the literature where such structures were previously unconnected or derived via different methods.
• Unified Framework for DSWs: The derived solutions are successfully applied to both classical and non-classical dispersive shock regimes. The authors demonstrate how the variational soliton can be integrated into Whitham modulation theory to solve for the structure of resonant dispersive shocks, including the determination of the Whitham shock velocity and resonant wavenumber.

Results
The theoretical predictions derived from the variational approach show excellent agreement with direct numerical simulations of the conservative-eKdV equation:

• Classical DSWs: Comparisons for the leading solitary wave edge velocity ($V_s$) and height ($A_s$) across a range of initial jump magnitudes ($\Delta$) confirm the accuracy of the variational solutions when using standard shallow water coefficients.
• Resonant DSWs (CDSW): In the non-classical regime, the theory accurately predicts the resonant wavenumber ($k_r$), the Whitham shock velocity ($U_s$), the resonant wave height ($A_r$), and the leading edge height ($A_s$). Maximum errors reported are approximately 19.95% for $U_s$ and 11.48% for $A_s$ in one parameter set, and lower (11.5% and 9.07%) in another, validating the effectiveness of the variational ansatz even in complex resonant scenarios.

Significance
The paper claims that the primary significance of this work lies in the simplicity and applicability of the derived variational solitonic solutions. Unlike previous methods that required complex algebraic manipulations or nonlocal transformations, the variational approach yields solutions that are readily applicable to practical problems in modulation theory. This facilitates the analysis of both classical and non-classical dispersive hydrodynamic phenomena, particularly in the rapidly growing field of non-convex dispersive hydrodynamics where solitary wave approximations are essential for modeling undular bores and dispersive shocks. The authors note that while the current analysis is restricted to the energy-conserving case ($c_2 = 2c_3$), the framework provides a foundation for future investigations into the full eKdV equation where energy conservation is not enforced.