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Small-amplitude fully localised solitary waves for the full-dispersion Kadomtsev--Petviashvili equation (1802.04823v1)

Published 13 Feb 2018 in math.AP

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' orlump' 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_22}{k_12} \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.

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