arXiv · 2101.00934
Using the nonabilinization procedure, we find an integrable matrix version of the Euler top on $\mathfrak{so}_3$
Explore connections, maps & timelines
V. Sokolov. 2021-01-04. Nonabelian ${\mathfrak so}_3$ Euler top. https://arxiv.org/abs/2101.00934
Cite the original work for its findings. Save a collection to share your selection of sources.
KEEP EXPLORING
A collection of miscellaneous continuous, semi-discrete, and discrete integrable systems can be associated with each integrable evolution equation of the KdV type. We give them for the Schwarz--KdV equation and generalize to the vector case. The existence of these vector generalizations is a non-trivial experimental fact for which no mathematical explanation is yet known.
Double ramification (DR) hierarchies associated to rank-$1$ F-CohFTs are important integrable perturbations of the Riemann--Hopf hierarchy. In this paper, we perform bihamiltonian tests for these DR hierarchies, and conjecture that the ones that are bihamiltonian form a $2$-parameter family. Remarkably, our computations suggest that there is a $1$-parameter subfamily of the rank-$1$ F-CohFTs, where the corresponding DR hierarchy is conjecturally Miura equivalent to the Camassa--Holm hierarchy. We also prove a conjecture regarding bihamiltonian Hodge hierarchies. Finally, we systematically study Miura invariants, and for another $1$-parameter subfamily propose a conjectural relation to the Degasperis--Procesi hierarchy.
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.