Small-amplitude fully localised solitary waves for the full-dispersion Kadomtsev–Petviashvili equation

Abstract

The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number ). This equation admits—as an explicit solution—a 'fully localised' or 'lump' solitary wave which decays to zero in all spatial directions. Recently there has been interest in the full-dispersion KP-I equation where is the Fourier multiplier with symbol which is obtained by retaining the exact dispersion relation from the water-wave problem. In this paper we show that the FDKP-I equation also has a fully localised solitary-wave solution. The existence theory is variational and perturbative in nature. A variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the variational functional associated with fully localised solitary-wave solutions of the KP-I equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.