On integrability of some Riemann type hydrodynamical systems and Dubrovin integrability classification of perturbed Korteweg-de Vries type equations

In our report we will stop on two closely related to each other integrability theory aspects. The first one concerns the obtained integrability results, based on the gradient-holonomic integrability scheme, devised and applied by me jointly with Maxim Pavlov and collaborators to a virtually new important Riemann type hierarchy $D_{t}^{N-1}u=z_{x}^{s}$, $D_{t}z=0$, where $s$, $N\in\mathbb{N}$ are arbitrary natural numbers, and proposed in our work (M. Pavlov, A. Prykarpatsky, at al., arXiv:1108.0878) as a nontrivial generalization of the infinite hierarchy of the Riemann type flows, suggested before by M. Pavlov and D. Holm in the form of dynamical systems $D_{t}^{N}u=0$, defined on a $2\pi$-periodic functional manifold $M^{N}\subset C^{\infty}(\mathbb{R}/2\pi\mathbb{Z};\mathbb{R}^{N})$, the vector $(u,D_{t}u,D_{t}^2u,...,D_{t}^{N-1}u,z)^{\intercal}\in M^{N}$, the differentiations $D_{x}:=\partial/\partial x$, $D_{t}:=\partial/\partial t+u\partial/\partial x$ satisfy as above the Lie-algebraic commutator relationship $[D_{x},D_{t}]=u_{x}D_{x}$ and $t\in\mathbb{R}$ is an evolution parameter. The second aspect of our report concerns the integrability results obtained by B. Dubrovin jointly with Y. Zhang and collaborators, devoted to classification of a special perturbation of the Korteweg-de Vries equation in the form $u_{t}=uu_{x}+\epsilon^2[f_{31}(u)u_{xxx}+f_{32}(u)u_{xx}u_{x}+f_{33}(u)u_{x}^3]$, where $f_{jk}(u),~j=3,~k=1,~3$, are some smooth functions and $\epsilon\in\mathbb{R}$ is a real parameter. We will deal with classification scheme of evolution equations of a special type suspicious on being integrable which was devised some years ago by untimely passed away Prof. Boris Dubrovin (19 March 2019) and developed with his collaborators, mainly with Youjin Zhang. We have reanalyzed in detail their interesting results on integrability classification of a suitably perturbed KdV type equation within our gradient-holonomic integrability scheme, devised many years ago and developed by me jointly with Maxim Pavlov and collaborators, and found out that the Dubrovin's scheme has missed at least a one very interesting integrable equation, whose natural reduction became similar to the well-known Krichever-Novikov equation, yet different from it. As a consequence of the analysis, we presented one can firmly claim that the Dubrovin-Zhang integrability criterion inherits some important part of the mentioned above gradient-holonomic integrability scheme properties, coinciding with the statement about the necessary existence of suitably ordered reduction expansions with coefficients to be strongly homogeneous differential polynomials.
Joint with Alex A. Balinsky, Radoslaw Kycia and Yarema A. Prykarpatsky.