Related papers: Dispersionless Hirota equations and the genus 3 hy…

We discuss some properties of the soliton equations of the type, partial derivative u/partial derivative t = S [u, (u) over bar], where S is a nonlinear operator differential in x, and present the additivity theorems of the class of the…

Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely…

We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include…

We constructed Hirota-Kimura type discretization of the classical nonholonomic Suslov problem of motion of rigid body fixed at a point. We found a first integral proving integrability. Also, we have shown that discrete trajectories…

We propose a new method based on the invariant properties of the ansatz of the Hirota method which have been discovered recently. This method allows one to construct new solutions for a certain class the dissipative equations classified by…

Let $M$ be an irreducible Hermitian symmetric space of compact type, and let $\omega$ be its K\"ahler form. For a triplet $(p_1,p_2,p_3)$ of points in $M$ we study conditions under which a geodesic triangle $\mathcal T(p_1,p_2,p_3)$ with…

We compute the class of the closure of the locus of hyperelliptic curves in the moduli space of stable genus-3 curves in terms of the tautological class $\lambda$ and the boundary classes $\delta_0$ and $\delta_1$. The expression of this…

The dispersionless differential Fay identity is shown to be equivalent to a kernel expansion providing a universal algebraic characterization and solution of the dispersionless Hirota equations. Some calculations based on D-bar data of the…

In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function, Baker-Akhiezer functions and "current amplitudes",…

Given a dihedral $2$-group $P$ of order at least~8, we classify the splendid Morita equivalence classes of principal $2$-blocks with defect groups isomorphic to $P$. To this end we construct explicit stable equivalences of Morita type…

We show that solution to the Hermite-Pad\'{e} type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the…

Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon - type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we…

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial…

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this…

We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the…

Spin-orbital liquids provide an exactly solvable route to three-dimensional Z2 quantum spin liquids beyond the original Kitaev setting. Built from higher-dimensional Clifford-algebra representations, spin-orbital Hamiltonians can be…

We classify integrable Hamiltonian equations in 3D with the Hamiltonian operator d/dx, where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality w such that w_x=u_y. Based on the method…

We solve the Killing spinor equations of 6-dimensional (1,0)-supergravity coupled to any number of tensor, vector and scalar multiplets in all cases. The isotropy groups of Killing spinors are $Sp(1)\cdot Sp(1)\ltimes \bH (1)$, $U(1)\cdot…

The relation between the infinite-dimensional 3-algebras and the dispersionless KdV hierarchy is investigated. Based on the infinite-dimensional 3-algebras, we derive two compatible Nambu Hamiltonian structures. Then the dispersionless KdV…

This is a review of the relationship between Fay identities and Hirota equations in integrable systems, reformulated in a geometric language compatible with recent Topological Recursion formalism. We write Hirota equations as trans-series,…