From infinite symmetries to multi-soliton manifolds: dispersion-driven reduction for the KP and mKP equations
Abstract
The infinite-dimensional symmetry algebras of integrable systems in (2+1) dimensions exhibit a remarkable reduction property when restricted to finite-dimensional multi-soliton manifolds. In this paper, we systematically investigate this phenomenon for the Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili equations. For the Kadomtsev-Petviashvili equation, we analyze both the K-symmetry and the τ-symmetry hierarchies. It is shown that for any n-soliton solution, each higher-order K-symmetry generator reduces to a linear combination of the n independent translational symmetries σξm0 associated with the individual soliton phase shifts, where the coefficients are precisely the dispersion relations evaluated at the soliton parameters. Closed-form expressions for odd- and even-order K-symmetries are derived, and determinant representations are provided to characterize the reduction systematically. For the τ-symmetries, each τn becomes a linear combination of the three fundamental soliton symmetries σkm, σlm, and σξm0, with coefficients that are rational functions of soliton parameters km and lm. These results are further extended to the modified Kadomtsev-Petviashvili equation, where analogous reduction patterns are established. This work establishes a direct and intrinsic link between the algebraic structure of infinite symmetries and the spectral properties of soliton solutions, offering a unified physical interpretation: the infinite set of higher-order symmetries generates linear superpositions of elementary soliton translations, weighted by powers of the wave numbers and dispersion coefficients.
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BibTeX
@misc{hao2026from,
title = {{From infinite symmetries to multi-soliton manifolds: dispersion-driven reduction for the KP and mKP equations}},
author = {Xiazhi Hao and Mingxia Ji and Senyue Lou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29770},
eprint = {2610.29770},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29770},
}
RIS
TY - UNPB
TI - From infinite symmetries to multi-soliton manifolds: dispersion-driven reduction for the KP and mKP equations
AU - Xiazhi Hao
AU - Mingxia Ji
AU - Senyue Lou
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29770
UR - https://arxiv.org/abs/2610.29770
AB - The infinite-dimensional symmetry algebras of integrable systems in (2+1) dimensions exhibit a remarkable reduction property when restricted to finite-dimensional multi-soliton manifolds. In this paper, we systematically investigate this phenomenon for the Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili equations. For the Kadomtsev-Petviashvili equation, we analyze both the K-symmetry and the τ-symmetry hierarchies. It is shown that for any n-soliton solution, each higher-order K-symmetry generator reduces to a linear combination of the n independent translational symmetries σ_ξ_m0 associated with the individual soliton phase shifts, where the coefficients are precisely the dispersion relations evaluated at the soliton parameters. Closed-form expressions for odd- and even-order K-symmetries are derived, and determinant representations are provided to characterize the reduction systematically. For the τ-symmetries, each τ_n becomes a linear combination of the three fundamental soliton symmetries σ_k_m, σ_l_m, and σ_ξ_m0, with coefficients that are rational functions of soliton parameters k_m and l_m. These results are further extended to the modified Kadomtsev-Petviashvili equation, where analogous reduction patterns are established. This work establishes a direct and intrinsic link between the algebraic structure of infinite symmetries and the spectral properties of soliton solutions, offering a unified physical interpretation: the infinite set of higher-order symmetries generates linear superpositions of elementary soliton translations, weighted by powers of the wave numbers and dispersion coefficients.
ER -