# Numerical study of the Kadomtsev--Petviashvili equation and dispersive shock waves

## TL;DR

This paper numerically investigates the long-term behavior of dispersive shock waves in the KP I equation, revealing the emergence of lump solutions and contrasting modulation dynamics between KPI and KPII.

## Contribution

It derives Whitham modulation equations for KP and analyzes their hyperbolic or elliptic nature, explaining lump formation and focusing effects.

## Key findings

- Lump solutions emerge from dispersive shock waves in KP.
- Modulation equations are hyperbolic for KPII and elliptic for KPI.
- Elliptic modulation leads to focusing and lump formation.

## Abstract

A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrodinger equation in the semiclassical limit.