A generalized long-wave limit method with spectral perturbations
Abstract
We develop a generalized long-wave limit method for constructing higher-order (degenerate) lump solutions of (2+1)-dimensional integrable systems. Starting from the multi-soliton tau function, we introduce wavenumber rescalings, spectral perturbations, and higher-order phase modulations in a single small parameter, and choose their coefficients so that the lower-order Taylor coefficients vanish. Applying this framework to the Kadomtsev--Petviashvili I equation, we explicitly derive second-order two- and three-lump solutions as well as third-order three-, four-, five-, and six-lump solutions. The long-time asymptotics of the peak positions and amplitudes describe anomalous scattering of these degenerate lump solutions. Extensions to hybrid higher-order lump solutions associated with multiple distinct spectral parameters are also discussed.
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BibTeX
@misc{qiu2026generalized,
title = {{A generalized long-wave limit method with spectral perturbations}},
author = {Tianwei Qiu and Zhen Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.15174},
eprint = {2606.15174},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.15174},
}
RIS
TY - UNPB
TI - A generalized long-wave limit method with spectral perturbations
AU - Tianwei Qiu
AU - Zhen Wang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2606.15174
UR - https://arxiv.org/abs/2606.15174
AB - We develop a generalized long-wave limit method for constructing higher-order (degenerate) lump solutions of (2+1)-dimensional integrable systems. Starting from the multi-soliton tau function, we introduce wavenumber rescalings, spectral perturbations, and higher-order phase modulations in a single small parameter, and choose their coefficients so that the lower-order Taylor coefficients vanish. Applying this framework to the Kadomtsev--Petviashvili I equation, we explicitly derive second-order two- and three-lump solutions as well as third-order three-, four-, five-, and six-lump solutions. The long-time asymptotics of the peak positions and amplitudes describe anomalous scattering of these degenerate lump solutions. Extensions to hybrid higher-order lump solutions associated with multiple distinct spectral parameters are also discussed.
ER -