Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations

TL;DR

This paper numerically investigates shock formation in the dispersionless KP equation, analyzing singularity development, regularizations, and the effects of small dispersion, with implications for understanding dispersive shock phenomena.

Contribution

It introduces a numerical approach to identify singularities in dKP solutions and compares dispersive and dissipative regularizations, revealing scaling laws near shock points.

Findings

Fourier coefficients effectively identify singularity formation.

Dispersive regularizations like KP maintain global regularity for small dispersion.

Solutions differ near critical points by approximately epsilon^{2/7}.

Abstract

The formation of singularities in solutions to the dispersionless Kadomtsev-Petviashvili (dKP) equation is studied numerically for different classes of initial data. The asymptotic behavior of the Fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity. The approach is first tested in detail in 1+1 dimensions for the known case of the Hopf equation, where it is shown that the break-up of the solution can be identified with prescribed accuracy. For dissipative regularizations of this shock formation as the Burgers' equation and for dispersive regularizations as the Korteweg-de Vries equation, the Fourier coefficients indicate as expected global regularity of the solutions. The Kadomtsev-Petviashvili (KP) equation can be seen as a dispersive regularization of the dKP equation. The behavior of KP solutions for small dispersion parameter $\epsilon\ll 1$ near a break-up of corresponding dKP solutions is studied. It is found that the difference between KP and dKP solutions for the same initial data at the critical point scales roughly as $\epsilon^{2/7}$ as for the Korteweg-de Vries equation.