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

Abstract

The KP-I equation (ut − 2uux + 1 2 (β − 1 3 )uxxx)x − uyy = 0 arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number β > 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 ut + m(D)ux + 2uux = 0, where m(D) is the Fourier multiplier with symbol m(k) = 1 + β|k| 2 | 1 2 tanh |k| |k| 1 2 1 + 2k 2 2 k 2 1 1 2, which is obtained by retaining the exact dispersion relation from the waterwave 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.

The Cauchy Problem for the Fractional Kadomtsev--Petviashvili Equations

“…Thus, similarly to the Whitham equation, (1.10) displays various regimes depending on the frequency range. In particular it is proven in [10] that (1.10) with β > 0 possesses solitary wave solutions close to the "lumps" of the KP-I equation. We plan to come back to those issues in a next paper.…”

On Whitham and Related Equations

“…On the other hand, the Whitham equation can be viewed as the one‐dimensional restriction of the full dispersion KP equation introduced in to overcome the “bad” behavior of the dispersion relation of the usual KP equations at low frequencies in x (see also the analysis in ). We refer to for a further study of the Cauchy problem and to for the existence of localized solitary waves, “close” to the usual KP I ones in the case of strong surface tension: with where is a dimensionless coefficient measuring the surface tension effects and …”

“…On the other hand the Whitham equation can be viewed as the onedimensional restriction of the full dispersion KP equation introduced in [40] to overcome the "bad" behavior of the dispersion relation of the usual KP equations at low frequencies in x (see also the analysis in [41]). We refer to [44] for a further study of the Cauchy problem and to [20] for the existence of localized solitary waves, "close" to the usual KP I ones in the case of strong surface tension):…”

Dispersive estimates for full dispersion KP equations

“…A similar obstruction holds for the FDKP equations. In particular, the localised solitary waves solutions found in [6] cannot decay fast at infinity. 1.2.…”