On critical behaviour in generalized Kadomtsev--Petviashvili equations

TL;DR

This paper conjectures an asymptotic description for dispersive shock wave formation in generalized KP equations, using Painleve' I hierarchy solutions, supported by numerical evidence.

Contribution

It introduces a novel asymptotic framework based on Painleve' I hierarchy for generalized KP equations, with numerical validation.

Findings

Asymptotic description matches numerical simulations

Painleve' I hierarchy solutions characterize shock formation

Strong numerical evidence supports the conjecture

Abstract

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev--Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painleve' I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture.