KdV hierarchies and quantum Novikov's equations
Abstract
The paper begins with a review of the well known Novikov's equations and corresponding finite KdV hierarchies. For a positive integer $N$ we give an explicit description of the $N$th Novikov's equation and its first integrals. Its finite KdV hierarchy consists of $N$ compatible integrable polynomial dynamical systems in $\mathbb{C}^{2N}$. Then we discuss a noncommutative version of the $N$th Novikov's equation defined on a finitely generated free associative algebra $\mathfrak{B}_N$ with $2N$ generators. In $\mathfrak{B}_N$, for $N=1,2,3,4$, we have found twosided homogeneous ideals $\mathfrak{Q}_N\subset\mathfrak{B}_N$ (quantisation ideals) which are invariant with respect to the $N$th Novikov's equation and such that the quotient algebra $\mathfrak{C}_N = \mathfrak{B}_N\diagup \mathfrak{Q}_N$ has a well defined PoincareBirkhoffWitt basis. It enables us to define the quantum $N$th Novikov's equation on the $\mathfrak{C}_N$. We have shown that the quantum $N$th Novikov's equation and its finite hierarchy can be written in the standard Heisenberg form.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.06357
 Bibcode:
 2021arXiv210906357B
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 Mathematics  Dynamical Systems;
 Mathematics  Quantum Algebra
 EPrint:
 A revised version of the original submission: we have improved some notations, corrected typos, extended the list of references and added a few intermediate statements. The final results stay unchanged