A plethora of fully localised solitary waves for the full-dispersion Kadomtsev-Petviashvili equation

TL;DR

This work constructs fully localised solitary waves for the full-dispersion KP-I equation by perturbing from the classical KPI lump solutions. Using a Lyapunov-Schmidt-type reduction and a low-regularity implicit-function theorem in anisotropic function spaces, the authors show that for each KPI lump index $k$ there exists a smooth FDKP-I solitary wave with speed $c=1-\varepsilon^2$ that remains close to the KPI lump in the KP scaling. The reduced problem converges to the KPI steady equation as $\varepsilon\to0$, enabling a rigorous perturbative existence result and detailed asymptotics $u_k^\star(x,y)\approx\varepsilon^2\zeta_k^\star(\varepsilon x,\varepsilon^2 y)$. This extends the family of fully localised waves to the full-dispersion model and provides a robust connection between KPI lumps and their full-dispersion solitary-wave counterparts, with precise functional-analytic control over the construction.

Abstract

The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with Bond number $β>1/3$, also called strong surface tension. This equation has recently been shown to have a family of nondegenerate, symmetric `fully localised' or `lump' solitary waves which decay to zero in all spatial directions. The full-dispersion KP-I equation is obtained by retaining the exact dispersion relation in the modelling from the water-wave problem. In this paper we show that the FDKP-I equation also has a family of symmetric fullly localised solitary waves which are obtained by casting it as a perturbation of the KP-I equation and applying a suitable variant of the implicit-function theorem.

Figures (3)

• Figure 1: The KP lumps $\zeta_1^\star$ (left) and $\zeta_2^\star$ (right)
• Figure 2: FKDP-I dispersion relation for two-dimensional wave trains
• Figure 3: The cone $C = \{ \mathbf{k} \in {\mathbb R}^2 \colon |k_1| \leq \delta, |\tfrac{k_2}{k_1}| \leq \delta\}$

Theorems & Definitions (40)

• Lemma 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Theorem 3.1