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

### Abstract

The KP-I equation $(u_t-2uu_x+\tfrac{1}{2}(\beta-\tfrac{1}{3})u_{xxx})_x -u_{yy}=0$ 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 \emph{full-dispersion KP-I equation} $u_t + m({\mathrm D}) u_x + 2 u u_x = 0,$ where $m({\mathrm D})$ is the Fourier multiplier with symbol $m(k) = \left( 1 + \beta |k|^2|\right)^{\frac{1}{2}} \left( \frac{\tanh |k|}{|k|} \right)^{\frac{1}{2}} \left( 1 + \frac{2k_2^2}{k_1^2} \right)^{\frac{1}{2}},$ 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.Comment: 31 page

Topics: Mathematics - Analysis of PDEs, 35Q53, 35A15, 76B15
Publisher: 'IOP Publishing'
Year: 2018