Solitary waves of moderate amplitude in the SSGGN equations: the extended KdV-Whitham approximation

This paper demonstrates that the extended KdV-Whitham approximation and the slow space formulation of the eKdV equation serve as effective regularizations for modeling moderately nonlinear surface water waves, successfully mitigating the spurious resonant radiation issues found in the standard eKdV model while maintaining close agreement with the parent SSGGN system.

The Gist

Imagine you are watching a giant wave roll across the ocean. Scientists have been trying to write a "rulebook" (a mathematical equation) to predict exactly how these waves move, how they crash, and how they interact with each other.

For a long time, the most famous rulebook was the KdV equation. Think of this as a "basic map" of the ocean. It works perfectly for small, gentle ripples, but if the waves get too big or too choppy, the map starts to show errors. It's like trying to navigate a stormy sea with a map designed for a calm lake.

To fix this, scientists created a more advanced map called the extended KdV (eKdV) equation. This new map includes more details about how big waves behave. However, the authors of this paper discovered a major glitch in this new map: it creates "ghost waves."

The Problem: The "Ghost Train"

When the scientists used the advanced eKdV map to simulate a big wave, the math started producing a strange, invisible train of tiny ripples racing ahead of the main wave. In the real ocean (and in their most accurate, complex computer models), these "ghost waves" don't exist. They are just an artifact—a bug in the math caused by the way the equation handles speed and distance.

It's like driving a car where the speedometer is slightly broken; the car is moving fine, but the dashboard tells you there's a phantom car speeding ahead of you, confusing your navigation.

The Solution: Two Ways to Fix the Map

The authors, a team of mathematicians, proposed two clever ways to fix this glitch so the map matches reality again.

1. Change the "Time Zone" (The Slow Space Approach)
Imagine you are watching a movie. Usually, you watch it second-by-second (slow time). But sometimes, if you look at the movie frame-by-frame from a different angle (slow space), the glitches disappear.
The authors showed that if you rewrite the eKdV equation using a different perspective (focusing on space rather than time), the "ghost waves" vanish. The map becomes accurate again without needing to change the core rules.

2. The "Whitham" Upgrade (The Hybrid Engine)
If you must use the original "slow time" perspective, the authors suggest a hybrid fix. They took the advanced eKdV equation and swapped out its "engine" (the part that handles wave speed and dispersion) with the engine from the most accurate, complex model they have (called the SSGGN system).
Think of it like taking a high-performance sports car (the eKdV) but swapping its transmission with one from a tank (the SSGGN) to make it handle rough terrain better. They call this the extended KdV–Whitham (eKdVW) approximation.

• The Result: This hybrid model is incredibly accurate. It captures the big, solitary waves perfectly and, crucially, it stops the "ghost waves" from appearing. It's like having a GPS that never loses signal, even in a storm.

How to Choose the Right Tool

The paper also gives a "cheat sheet" for scientists on which map to use before they even start running their simulations.

• If the wave is going to stay mostly as one big lump: The standard advanced map (eKdV) is fine.
• If the wave is going to break apart into many smaller ripples: You must use the upgraded hybrid map (eKdVW).

They figured this out by looking at the "energy budget" of the wave. If the wave has a lot of energy that will turn into ripples, the basic map will fail, and the hybrid map is the only way to get it right.

The Bottom Line

This paper is about cleaning up the math we use to predict ocean waves.

• The Old Way: Good for small waves, but creates fake "ghost waves" for big ones.
• The New Way: By either changing the perspective or mixing the best parts of different equations, the authors created a model that works for moderate-sized waves (the kind we actually see in the ocean) without the annoying glitches.

It's a bit like upgrading from a paper map to a live satellite feed: the view is clearer, the errors are gone, and you can finally trust where the wave is going to go.

Technical Summary

1. Problem Statement

The paper addresses the modeling of moderately nonlinear surface water waves. While the classical Korteweg–de Vries (KdV) equation is a standard model for weakly nonlinear, weakly dispersive waves, its accuracy deteriorates as the wave amplitude increases (moderate nonlinearity). To address this, the extended KdV (eKdV) equation is often used, which includes higher-order nonlinear and dispersive terms.

However, the authors identify a critical flaw in the standard slow-time formulation of the eKdV equation:

• Resonant Radiation: Due to the non-convexity of its linear dispersion curve, the slow-time eKdV equation generates unphysical resonant radiation (a modulated wavetrain) ahead of the main solitary wave.
• Discrepancy with Parent Systems: This radiation does not exist in the fully nonlinear parent system, the 1D Serre–Su–Gardner–Green–Naghdi (SSGGN) equations, nor in the slow-space formulation of the eKdV equation.
• Goal: The study aims to regularize the eKdV equation to eliminate this spurious radiation while maintaining accuracy for moderate amplitudes, and to provide a method for selecting the most appropriate reduced-order model based on initial conditions.

2. Methodology

A. Derivation of Models

The authors derive the eKdV equation in two different coordinate frames from the 1D SSGGN equations using asymptotic multiple-scale expansions:

1. Slow-Time Formulation ($T = \epsilon t, \xi = x - t$): The standard derivation where resonant radiation occurs.
2. Slow-Space Formulation ($X = \epsilon x, \xi = x - t$): A variation where the linear dispersion relation remains convex, theoretically preventing resonance.

They also derive coefficients for the eKdV equation based on both the full SSGGN system and the simplified Boussinesq–Peregrine (BP) equations, noting differences in high-order coefficients.

B. The Extended KdV–Whitham (eKdVW) Approximation

To regularize the slow-time eKdV equation, the authors introduce the eKdVW approximation. This method:

• Retains the nonlinear terms of the eKdV equation (up to second order).
• Replaces the linear dispersive operator with the exact linear dispersion relation of the parent SSGGN system (via a non-local integral kernel).
• This creates an integro-differential equation that preserves the nonlinearity of the reduced model while correcting the dispersion to match the physical parent system.

C. Asymptotic Solutions (NIT)

To generate accurate initial conditions for numerical simulations, the authors construct solitary wave solutions for the eKdV equation using a Kodama-Fokas-Liu Near-Identity Transformation (NIT).

• They map the non-integrable eKdV equation to the integrable KdV equation.
• This yields an asymptotic solitary wave solution accurate to $O(\epsilon^2)$.
• They compare this against solutions derived from the "Improved Gardner" equation, finding the KdV-based NIT superior for surface waves where the quadratic nonlinearity is significant.

D. Numerical Simulations

The authors perform direct numerical simulations comparing:

• Parent Model: 1D SSGGN equations (solved via pseudospectral methods and GMRES for elliptic inversion).
• Reduced Models: KdV, eKdV (slow-time), eKdVW, Truncated Gardner, and Improved Gardner.
• Scenarios: Positive amplitude solitary waves, negative amplitude dispersive wave trains, and initial conditions evolving into either pure solitons or dispersive radiation.

3. Key Contributions

1. Identification of Resonance Source: The paper rigorously demonstrates that the resonant radiation in the slow-time eKdV equation is an artifact of its non-convex dispersion relation, which is absent in both the parent SSGGN system and the slow-space eKdV formulation.
2. Introduction of eKdVW: The authors propose the extended KdV–Whitham (eKdVW) approximation as an efficient regularization of the eKdV equation. It eliminates spurious radiation while retaining the higher-order accuracy of the eKdV nonlinear terms.
3. Predictive Model Selection: The study introduces a method to predict the best reduced model a priori using the Inverse Scattering Transform (IST) on the initial condition. By calculating the conserved quantities (mass, momentum, energy) of the KdV equation and comparing them to the initial state, one can determine the amount of energy destined for radiation.
   • Low Radiation: eKdV (slow-time) performs well.
   • High Radiation: eKdVW (or KdVW) is required for accuracy.
4. Comparative Analysis of Formulations: The paper provides a comprehensive comparison showing that the slow-space formulation is a viable alternative to the slow-time formulation for avoiding resonance, though the eKdVW offers a robust solution within the standard slow-time framework.

4. Key Results

• Positive Amplitude Solitons:

  • The standard eKdV equation produces a modulated resonant wavetrain ahead of the soliton, which grows with the amplitude parameter $\epsilon$.
  • The eKdVW equation eliminates this radiation entirely, showing excellent agreement with the SSGGN parent system in both amplitude and phase, even for moderate $\epsilon$ (e.g., $\epsilon = 0.4$).
  • Gardner-type equations (truncated and improved) perform worse than eKdV/eKdVW, exhibiting incorrect speeds or amplitudes.

• Negative Amplitude (Dispersive) Waves:

  • For negative initial conditions (which do not form solitons in surface water), the eKdV equation fails significantly, with resonance destroying the wave structure.
  • The eKdVW equation is indistinguishable from the parent SSGGN system in these scenarios, proving its superiority for dispersive wave trains.

• Slow-Space vs. Slow-Time:

  • Simulations in the slow-space variable ($X$) confirm that resonance is absent, validating the theoretical dispersion analysis. However, the eKdVW allows the use of the standard slow-time variable while achieving similar accuracy.

• IST-Based Prediction:

  • Numerical tests confirmed that when the initial condition leads to pure solitons (zero radiation in KdV IST analysis), the eKdV equation is the most accurate.
  • When the initial condition leads to significant radiation, the eKdVW equation is the only reduced model that maintains high accuracy.

5. Significance

This work is significant for the field of water wave modeling and nonlinear dispersive wave theory for several reasons:

• Resolution of a Long-standing Issue: It provides a practical and efficient solution to the problem of spurious resonant radiation in higher-order KdV models, a known issue that limits the utility of the eKdV equation for moderate amplitudes.
• Computational Efficiency: The eKdVW approximation is computationally cheaper than solving the full SSGGN equations (which require solving elliptic PDEs) while offering comparable accuracy for a wide range of moderate nonlinearities.
• Universal Applicability: The methodology (Whitham regularization and IST-based prediction) is not limited to surface water waves; it can be applied to other physical contexts where reduced-order models suffer from dispersion inaccuracies (e.g., internal waves, plasma physics, or waves in solids).
• Practical Guidance: The paper offers a clear, quantitative criterion (via conserved quantities) for researchers to choose the correct reduced model before running expensive simulations, preventing the use of inappropriate models for specific initial conditions.