Numerical study of the Kadomtsev--Petviashvili equation and dispersive shock waves. (arXiv:1706.04104v1 [math.NA])

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 of the line soliton 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 behavior is similar to breather appearance for the focusing nonlinear Schr\"odinger equation in the semiclassical limit. 查看全文>>