Modulation theory of Y- and X-wave Kadomtsev--Petviashvili solitons
Abstract
The evolution of long wavelength perturbations of traveling wave 2-soliton solutions of the Kadomtsev-Petviashvili II (KP) equation are investigated using modulation theory. An initial-boundary value problem for a system of hyperbolic conservation laws is formulated for amplitude and slope perturbations to soliton legs of the resonant (Y-shaped) and X-shaped KP traveling wave solutions. The vertex where individuated soliton legs coalesce is modeled as a moving free boundary where information between the soliton legs can pass. The linearized hyperbolic system is proven to be well-posed under suitable initial conditions and is solved exactly. In the case of the resonant Y-shaped soliton, all admissible simple (rarefaction) waves of the nonlinear system are characterized. Specific perturbations of Y-waves can lead to the dimensionality of the modulated solution changing, corresponding to a physical geometric change to the multi-soliton pattern. Spontaneous leg deletion and leg creation can occur. The conditions under which the Y-wave modulation system can experience gradient catastrophe are given. Gradient catastrophe is regularized by the formation of a new leg, with an augmented modulation system proposed to describe the growth of the new leg. On the other hand, the X-wave system can be reduced to two independent line soliton modulation systems. These findings reinforce the prevalence and generality of resonant Y-shaped vertices in the KP II system and applications.The theoretical modulation predictions are supported by direct numerical simulations of KP II, which show good agreement with the analysis.
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BibTeX
@misc{wang2026modulation,
title = {{Modulation theory of Y- and X-wave Kadomtsev--Petviashvili solitons}},
author = {Jerry Wang and Mark A. Hoefer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31425},
eprint = {2610.31425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31425},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Modulation theory of Y- and X-wave Kadomtsev--Petviashvili solitons
AU - Jerry Wang
AU - Mark A. Hoefer
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31425
UR - https://arxiv.org/abs/2610.31425
AB - The evolution of long wavelength perturbations of traveling wave 2-soliton solutions of the Kadomtsev-Petviashvili II (KP) equation are investigated using modulation theory. An initial-boundary value problem for a system of hyperbolic conservation laws is formulated for amplitude and slope perturbations to soliton legs of the resonant (Y-shaped) and X-shaped KP traveling wave solutions. The vertex where individuated soliton legs coalesce is modeled as a moving free boundary where information between the soliton legs can pass. The linearized hyperbolic system is proven to be well-posed under suitable initial conditions and is solved exactly. In the case of the resonant Y-shaped soliton, all admissible simple (rarefaction) waves of the nonlinear system are characterized. Specific perturbations of Y-waves can lead to the dimensionality of the modulated solution changing, corresponding to a physical geometric change to the multi-soliton pattern. Spontaneous leg deletion and leg creation can occur. The conditions under which the Y-wave modulation system can experience gradient catastrophe are given. Gradient catastrophe is regularized by the formation of a new leg, with an augmented modulation system proposed to describe the growth of the new leg. On the other hand, the X-wave system can be reduced to two independent line soliton modulation systems. These findings reinforce the prevalence and generality of resonant Y-shaped vertices in the KP II system and applications.The theoretical modulation predictions are supported by direct numerical simulations of KP II, which show good agreement with the analysis.
ER -