On the solutions of the dKP equation: nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

TL;DR

This paper investigates the dispersionless KP equation using a nonlinear Riemann Hilbert problem approach to analyze long-term solution behavior, implicit solutions, and the wave breaking mechanism causing gradient catastrophe.

Contribution

It introduces a novel application of the nonlinear Riemann Hilbert problem to characterize long-time behavior, implicit solutions, and wave breaking in the dKP equation.

Findings

Characterization of long-time behavior of solutions

Identification of implicit solution classes

Explanation of the spectral mechanism for wave breaking

Abstract

We make use of the nonlinear Riemann Hilbert problem of the dispersionless Kadomtsev Petviashvili equation, i) to construct the longtime behaviour of the solutions of its Cauchy problem; ii) to characterize a class of implicit solutions; iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking.