Solitary wave solutions to the Isobe-Kakinuma model for water waves

TL;DR

This work analyzes solitary-wave solutions in the Isobe--Kakinuma model for two-dimensional water waves with a flat bottom. It combines a formal long-wave $\delta$-expansion to derive leading-order solitary-wave profiles with $\eta_{(0)}(x)=4\gamma\sech^2x$ and $c=1+2\gamma\delta^2$, with a rigorous reduction to a linear remainder problem and a Green's-function framework that yields a contraction-based existence proof for small-amplitude solitary waves. It also provides a detailed numerical study in the $N=1$, $p_1=2$ case, revealing a maximum amplitude at $\delta_c\approx0.62633493$ and a solitary wave of extreme form with crest angle approximately $152.6^\circ$; this supports the model’s capability to capture crest formation beyond classical Green--Naghdi or KdV limits. Overall, the paper advances both the analytical construction and numerical characterization of solitary waves in a higher-order shallow-water approximation, highlighting the Isobe--Kakinuma model’s richer solitary-wave behavior.

Abstract

We consider the Isobe-Kakinuma model for two-dimensional water waves in the case of the flat bottom. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe-Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.

Figures (6)

• Figure 1.1: Solitary wave of extreme form with the included angle $\theta=\ang{152.6}$.
• Figure 7.1: Surface profiles of the solitary wave solutions to the Isobe--Kakinuma model (solid line) and the Korteweg--de Vries equation (dashed line) for several values of $\delta$: (a) $\delta=0.3$, (b) $\delta=0.45$, (c) $\delta=0.55$, (d) $\delta=0.6$, (e) $\delta=0.62$, (f) $\delta=0.62633493$.
• Figure 7.2: Surface profiles of the solitary wave solutions to the Isobe--Kakinuma model for $\delta=0.45$, 0.55, 0.6, 0.62, 0.62633493.
• Figure 7.3: Surface profiles near the crests of the solitary wave solutions to the Isobe--Kakinuma model for $\delta=0.6$, 0.62, 0.625, 0.626, 0.62633493.
• Figure 7.4: Surface profile of the solitary wave of extreme form to the Isobe--Kakinuma model.

Theorems & Definitions (7)

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