Oblique interactions between solitons and mean flows in the Kadomtsev–Petviashvili equation

Abstract

Abstract The interaction of an oblique line soliton with a one-dimensional dynamic mean flow is analyzed using the Kadomtsev–Petviashvili II (KPII) equation. Building upon previous studies that examined the transmission or trapping of a soliton by a slowly varying rarefaction or oscillatory dispersive shock wave (DSW) in one space and one time dimension, this paper allows for the incident soliton to approach the changing mean flow at a nonzero oblique angle. By deriving invariant quantities of the soliton–mean flow modulation equations—a system of three (1 + 1)-dimensional quasilinear, hyperbolic equations for the soliton and mean flow parameters—and positing the initial configuration as a Riemann problem in the modulation variables, it is possible to derive quantitative predictions regarding the evolution of the line soliton within the mean flow. It is found that the interaction between an oblique soliton and a changing mean flow leads to several novel features not observed in the (1 + 1)-dimensional reduced problem. Many of these interesting dynamics arise from the unique structure of the modulation equations that are nonstrictly hyperbolic, including a well-defined multivalued solution interpreted as a solution of the (2 + 1)-dimensional soliton–mean modulation equations, in which the soliton interacts with the mean flow and then wraps around to interact with it again. Finally, it is shown that the oblique interactions between solitons and DSW solutions for the mean flow give rise to all three possible types of two-soliton solutions of the KPII equation. The analytical findings are quantitatively supported by direct numerical simulations.