Poisson 铅笔、Lie 对称性与耦合非线性波系统的 Hamilton 约化：切 KdV 几何与椭圆模量
Poisson Pencils, Lie Symmetries and Hamiltonian Reductions of a Coupled Nonlinear Wave System:Tangent KdV Geometry and Elliptic Moduli
- Department of Mathematics and Statistics, Universidad Nacional de Colombia, Sede Manizales（哥伦比亚国立大学马尼萨莱斯校区数学与统计系）
- FIZMAKO Research Group, Manizales, Colombia（哥伦比亚马尼萨莱斯 FIZMAKO 研究组）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究双场非线性色散系统，将其作为 KdV 的切覆盖和 Hamilton 流，证明 Magri 铅笔的切提升、识别对称代数，并通过行波约化得到四维系统，在椭圆轨迹上显式分类切行波。
AI中文摘要：
我们研究双场非线性色散系统 \\[ u_t=u_{xxx}+6uu_x,\qquad v_t=v_{xxx}+6(uv)_x, \\]，该系统同时被视为耦合波动方程、Korteweg--de Vries (KdV) 方程的切覆盖，以及切 Poisson 流形上的 Hamilton 流。主要目的是在定理层面使这些观点相互作用。首先，我们证明 KdV 的 Magri Poisson 铅笔允许完全切提升为显式相容的矩阵 Hamilton 算子对。相应的递推算子具有三角切形式，并生成提升的 KdV 层次。其次，我们识别出一个五维点对称代数以及切广义对称性的无限层次。第三，通过行波子群约化产生一个四维 Hamilton 系统，它是标量 KdV 剖面动力学的完全切提升。在非奇异椭圆轨迹上，基础剖面以 Weierstrass 形式写出，每个切行波通过关于能量和积分常数的导数被显式分类。
英文摘要：
We study the two-field nonlinear dispersive system \[ u_t=u_{xxx}+6uu_x,\qquad v_t=v_{xxx}+6(uv)_x, \] viewed simultaneously as a coupled wave equation, as the tangent covering of Korteweg--de Vries (KdV), and as a Hamiltonian flow on a tangent Poisson manifold. The main purpose is to make these viewpoints interact at theorem level. First, we prove that the Magri Poisson pencil of KdV admits a complete tangent lift to an explicit compatible pair of matrix Hamiltonian operators. The corresponding recursion operator has triangular tangent form and generates the lifted KdV hierarchy. Second, we identify a five-dimensional point-symmetry algebra together with the infinite hierarchy of tangent generalized symmetries. Third, reduction by the traveling-wave subgroup produces a four-dimensional Hamiltonian system that is the complete tangent lift of the scalar KdV profile dynamics. On the nonsingular elliptic locus, the base profile is written in Weierstrass form and every tangent traveling wave is classified explicitly by derivatives with respect to the energy and integration constants.
