Deformations of (2+1)-dimensional integrable systems and isotropic and anisotropic Harry Dym variants
Abstract
The deformation algorithm based on conservation laws can lift (1+1)-dimensional integrable systems to higher-dimensional counterparts while preserving Lax integrability, yet its generalization to (2+1)-dimensional models has remained an open problem. This paper establishes a unified deformation framework for two (2+1)-dimensional integrable equations: the anisotropic Kadomtsev-Petviashvili (KP) equation and the isotropic Nizhnik-Novikov-Veselov (NNV) equation. By introducing a set of mutually commuting field-dependent deformation operators, we systematically construct infinite families of (m+3)-dimensional integrable KP and NNV hierarchies, derive their closed-form Lax pairs, and rigorously verify integrability via the vanishing commutator condition of Lax operators. For each high-dimensional hierarchy, concrete finite-dimensional master systems are obtained by truncating auxiliary spatial variables: a (3+1)-dimensional KP system and a (4+1)-dimensional generalized NNV system. Further symmetry reductions recover the original (2+1)-dimensional KP and NNV equations, and more importantly produce two distinct Harry-Dym (HD)-type reciprocal integrable systems. The KP reduction yields an anisotropic (2+1)-dimensional HD model, while the NNV reduction generates the first spatially isotropic two-space-dimensional HD system reported so far, filling a notable gap in existing literature. Parallel comparison of the KP and NNV branches reveals that the spatial symmetry of the original two-dimensional parent equation directly governs the symmetry properties of its high-dimensional deformations and HD dual subsystems. Our work not only extends the conservation-law deformation conjecture beyond (1+1)-dimensions to accommodate both strong Lax and weak Lax structures, but also provides a universal route to construct reciprocal links for multi-dimensional integrable systems.
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BibTeX
@misc{faren2026deformations,
title = {{Deformations of (2+1)-dimensional integrable systems and isotropic and anisotropic Harry Dym variants}},
author = {Wang Fa-Ren and Jia Man and Lou S Y},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.01293},
eprint = {2608.01293},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01293},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Deformations of (2+1)-dimensional integrable systems and isotropic and anisotropic Harry Dym variants
AU - Wang Fa-Ren
AU - Jia Man
AU - Lou S Y
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2608.01293
UR - https://arxiv.org/abs/2608.01293
AB - The deformation algorithm based on conservation laws can lift (1+1)-dimensional integrable systems to higher-dimensional counterparts while preserving Lax integrability, yet its generalization to (2+1)-dimensional models has remained an open problem. This paper establishes a unified deformation framework for two (2+1)-dimensional integrable equations: the anisotropic Kadomtsev-Petviashvili (KP) equation and the isotropic Nizhnik-Novikov-Veselov (NNV) equation. By introducing a set of mutually commuting field-dependent deformation operators, we systematically construct infinite families of (m+3)-dimensional integrable KP and NNV hierarchies, derive their closed-form Lax pairs, and rigorously verify integrability via the vanishing commutator condition of Lax operators. For each high-dimensional hierarchy, concrete finite-dimensional master systems are obtained by truncating auxiliary spatial variables: a (3+1)-dimensional KP system and a (4+1)-dimensional generalized NNV system. Further symmetry reductions recover the original (2+1)-dimensional KP and NNV equations, and more importantly produce two distinct Harry-Dym (HD)-type reciprocal integrable systems. The KP reduction yields an anisotropic (2+1)-dimensional HD model, while the NNV reduction generates the first spatially isotropic two-space-dimensional HD system reported so far, filling a notable gap in existing literature. Parallel comparison of the KP and NNV branches reveals that the spatial symmetry of the original two-dimensional parent equation directly governs the symmetry properties of its high-dimensional deformations and HD dual subsystems. Our work not only extends the conservation-law deformation conjecture beyond (1+1)-dimensions to accommodate both strong Lax and weak Lax structures, but also provides a universal route to construct reciprocal links for multi-dimensional integrable systems.
ER -