Existence of Davey--Stewartson type solitary waves for the fully dispersive Kadomtsev--Petviashvilii equation

TL;DR

This paper proves the existence of small-amplitude solitary waves for the fully dispersive Kadomtsev--Petviashvilii equation, showing they are perturbations of solutions to a Davey--Stewartson type equation, using a variational approach.

Contribution

It introduces a variational method to establish the existence and convergence of solitary waves for the FDKP equation, linking them to DS ground states, and demonstrates the method's robustness for similar dispersive equations.

Findings

Existence of small-amplitude solitary waves for FDKP.

Convergence of scaled FDKP waves to DS ground states.

Method applicable to other nonlinear dispersive equations with specific dispersion properties.

Abstract

We prove existence of small-amplitude modulated solitary waves for the full-dispersion Kadomtsev--Petviashvilii (FDKP) equation with weak surface tension. The resulting waves are small-order perturbations of scaled, translated and frequency-shifted solutions of a Davey--Stewartson (DS) type equation. The construction is variational and relies upon a series of reductive steps which transform the FDKP functional to a perturbed scaling of the DS functional, for which least-energy ground states are found. We also establish a convergence result showing that scalings of FDKP solitary waves converge to ground states of the DS functional as the scaling parameter tends to zero. Our method is robust and applies to nonlinear dispersive equations with the properties that (i) their dispersion relation has a global minimum (or maximum) at a non-zero wave number, and (ii) the associated formal weakly nonlinear analysis leads to a DS equation of elliptic-elliptic focussing type. We present full details for the FDKP equation.