发表机构
Osaka Metropolitan University; Universidade Federal de São Carlos (UFSCar)（大阪公立大学; 圣卡洛斯联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造圈代数中的Lax算子，得到Camassa-Holm方程的$\mathbb{Z}_2^2$-分次可积推广，并证明其具有双哈密顿结构及无穷多对合守恒量。
AI 中文摘要
通过在Lie超代数$\mathfrak{osp}(1|2)$的$\mathbb{Z}_2^2$-分次推广的圈代数中构造Lax算子，我们推导出Camassa-Holm方程的$\mathbb{Z}_2^2$-分次推广。所得方程是一个可积非线性偏微分方程，描述四个$\mathbb{Z}_2^2$-分次交换函数组成的系统，每个函数对应不同的$\mathbb{Z}_2^2$-次数。我们进一步证明该$\mathbb{Z}_2^2$-Camassa-Holm方程具有双哈密顿结构。因此，它拥有无穷多个守恒量，其中包括一个具有非平凡$\mathbb{Z}_2^2$-次数的守恒量，这些守恒量关于$\mathbb{Z}_2^2$-分次泊松括号两两对合。
英文摘要
By constructing Lax operators in the loop algebra of the $\mathbb{Z}_2^2$-graded extension of the Lie superalgebra $\mathfrak{osp}(1|2)$, we derive a $\mathbb{Z}_2^2$-graded extension of the Camassa-Holm equation. The resulting equation is an integrable nonlinear PDE for a system of four $\mathbb{Z}_2^2$-graded commutative functions, each associated with a distinct $\mathbb{Z}_2^2$-degree. We further show that the $\mathbb{Z}_2^2$-Camassa-Holm equation admits a bi-Hamiltonian structure. As a consequence, it possesses infinitely many conserved quantities, including one with non-trivial $\mathbb{Z}_2^2$-degree, which are mutually in involution with respect to the $\mathbb{Z}_2^2$-graded Poisson brackets.
Comments26 pages, no figure
