Half-horseshoe soliton of the KP-I equation

Abstract Since Ohta’s systematic summary of the Airy solitons for the KP-I equation in 2025 [Water Waves (2025) 7:349–362], the explicit analytical construction of such solutions has become essentially a closed topic, except for a few remaining intricate integral issues. Building on Ohta’s work, this brief study identifies a class of half-horseshoe solitons, whose crests are distributed along parabolic fronts, with amplitudes growing from infinitesimal to infinite. However, the resonant solutions generated by these line-like or lump-like half-horseshoe solitons have amplitudes that remain bounded in space. Although these solutions exhibit intriguing structures, the corresponding transformed cylindrical KP solutions still lack physical relevance, as they are non-periodic with respect to the azimuth.