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 Schrödinger equation in the semiclassical limit.
|Number of pages||20|
|Journal||Proceedings of the Royal Society A: Mathematical and Physical Sciences|
|Early online date||14 Feb 2018|
|Publication status||Published - 28 Feb 2018|
- Dispersive Shock waves
- Kadomtsev Petviashvili equatuon