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

TL;DR

The paper addresses moderately nonlinear surface waves by juxtaposing the extended KdV (eKdV) model with its slow-time and slow-space formulations against the 1D SSGGN parent system. A Whitham-based regularization is proposed to mitigate resonant radiation inherent in the slow-time eKdV, yielding the extended KdV–Whitham (eKdVW) model, while the slow-space formulation inherently avoids resonance. Through Kodama–Fokas–Liu near-identity transformations, the authors construct $O(\epsilon^2)$ accurate solitary-wave approximations and compare them to exact SSGGN solitons and Gardner-type reductions. Numerical experiments reveal that eKdV in slow time can exhibit resonant fronts, whereas slow-space regularization and especially the eKdVW model provide superior agreement with the parent system, particularly when radiative content is high or for negative amplitudes. An IST-based, conservation-law framework is proposed to predict the most suitable reduced model a priori, highlighting practical implications for efficiently simulating moderately nonlinear surface waves.

Abstract

We consider the extended Korteweg-de Vries (eKdV) equation as a model for long moderately nonlinear surface water waves. In the slow time formulation this equation generates fast propagating resonant radiation due to the non-convexity of its linear dispersion curve, which is not present in the strongly nonlinear Serre-Su-Gardner-Green-Naghdi (SSGGN) parent system. We show that the extended KdV-Whitham approximation and the slow space formulation of the eKdV equation are suitable regularisations of the eKdV equation in several cases of interest, and even for moderate amplitudes. Numerical comparisons are made between the SSGGN system and the respective reduced models, where simulations are initiated with an approximate soliton solution of the eKdV equation, constructed by use of Kodama-Fokas-Liu near-identity transformation to the KdV equation.

Figures (8)

• Figure 1: The phase velocity (a) and group velocity (b) for $\epsilon = 0.1$ of the KdV equation (blue), eKdV equation in slow time $T$ (red), eKdV equation in slow space $X$ (green), and the parent 1D SSGGN equations (black).
• Figure 2: Numerical solution of the eKdV equation for $\epsilon = 0.3$ where the solution is plotted at $T = 1$, $T = 5$, and $T = 15$.
• Figure 3: Analytical plots of the exact 1D SSGGN, KdV, and improved Gardner soliton solutions and the respective NIT solutions are plotted explicitly in ($\textit{a}$) $\epsilon = 0.1$ and ($\textit{b}$) $\epsilon = 0.4$ for $V = 0.5$. In ($\textit{c}$) and ($\textit{d}$) the $L_{\infty}$ norm of $\epsilon \eta$ and $\epsilon \Delta$ is plotted against $\epsilon V$, respectively, where $\Delta$ is the difference between the 1D SSGGN solution and respective reduced order solution.
• Figure 4: Numerical solutions of the 1D SSGGN equations and the corresponding reduced amplitude models where the left column gives the amplitude comparison and the right column gives the $L_{\infty}$ norm of the difference, $\Delta$, between the parent model and reduced model. Solutions are plotted at $T = 10$ for $V = 0.5$ where (a) $\epsilon = 0.1$, (b) $\epsilon = 0.2$, and (c) $\epsilon = 0.4$.
• Figure 5: Numerical solutions the 1D SSGGN equations and the corresponding reduced amplitude models where the left column gives the amplitude comparison and the right column gives the $L_{\infty}$ norm of the difference, $\Delta$, between the parent model and reduced model. Solutions are plotted at $T = 10$ for $V = 0.5$ where (a) $\epsilon = 0.1$ and (b) $\epsilon = 0.2$.