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

The KP-I equation (u(t) - 2uu(x) + 1/2 (beta - 1/3)u(xxx))(x) - u(yy) = 0 arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number beta > 1/3). 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 u(t) + m(D)u(x) + 2uu(x) = 0, where m(D) is the Fourier multiplier with symbol m(k) = (1 + beta vertical bar k vertical bar(2)vertical bar)(1/2) (tanh vertical bar k vertical bar/vertical bar k vertical bar)(1/2) (1 + 2k(2)(2)/k(1)(2))(1/2), 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.