发表机构
School of Physical Science and Technology, Ningbo University; Institute of Fundamental Physics and Quantum Technology, Ningbo University（宁波大学物理科学与技术学院; 宁波大学基础物理与量子技术研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立统一变形框架，将(2+1)维KP、NNV可积方程变形为高维可积系统，填补空间各向同性HD系统空白，为多维可积系统构建互易关联提供通用途径。
AI 中文摘要
基于守恒律的变形算法可将(1+1)维可积系统提升为高维对应系统，同时保持Lax可积性，但其向(2+1)维模型的推广仍为开放问题。本文为两个(2+1)维可积方程建立统一变形框架：各向异性Kadomtsev-Petviashvili（KP）方程与各向同性Nizhnik-Novikov-Veselov（NNV）方程。通过引入一组相互对易的场依赖变形算子，系统构造了无穷多族(m+3)维可积KP和NNV层级，推导其闭式Lax对，并通过Lax算子的对易子消失条件严格验证可积性。对每个高维层级，通过截断辅助空间变量得到具体有限维主系统：(3+1)维KP系统与(4+1)维广义NNV系统。进一步的对称约化不仅恢复原始(2+1)维KP和NNV方程，更产生两个不同的Harry-Dym（HD）型互易可积系统：KP约化得到各向异性(2+1)维HD模型，NNV约化生成迄今报道的首个空间各向同性两空间维HD系统，填补了现有文献的显著空白。对KP与NNV分支的平行比较表明，原始二维母方程的空间对称性直接决定其高维变形及HD对偶子系统的对称性质。本工作不仅将守恒律变形猜想从(1+1)维扩展至可容纳强Lax与弱Lax结构的情形，还为构造多维可积系统的互易关联提供了通用途径。
英文摘要
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.
Comments14 pages
