Resonant Yshaped soliton solutions to the Kadomtsev–Petviashvili II (KPII) equation are modelled as shock solutions to an infinite family of modulation conservation laws. The fully twodimensional soliton modulation equations, valid in the zero dispersion limit of the KPII equation, are demonstrated to reduce to a onedimensional system. In this same limit, the rapid transition from the larger Y soliton stem to the two smaller legs limits to a travelling discontinuity. This discontinuity is a multivalued, weak solution satisfying modified Rankine–Hugoniot jump conditions for the onedimensional modulation equations. These results are applied to analytically describe the dynamics of the Mach reflection problem, Vshaped initial conditions that correspond to a soliton incident upon an inward oblique corner. Modulation theory results show excellent agreement with direct KPII numerical simulation.
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Oblique interactions between solitons and mean flows in the Kadomtsev–Petviashvili equation
Abstract The interaction of an oblique line soliton with a onedimensional 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 welldefined 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 twosoliton solutions of the KPII equation. The analytical findings are quantitatively supported by direct numerical simulations.
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 Award ID(s):
 2009487
 NSFPAR ID:
 10318534
 Date Published:
 Journal Name:
 Nonlinearity
 Volume:
 34
 Issue:
 6
 ISSN:
 09517715
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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