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Register a new userGeometry of the Kahan discretizations of planar quadratic Hamiltonian systems
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Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic Hamiltonian vector field.
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Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper `Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation by P. van der Kamp et al., it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form $ell(x,y)$, let $B_1,B_2$ be any two distinct points on the line $ell(x,y)=-c$, and let $B_3,B_4$ be any two distinct points on the line $ell(x,y)=c$. Set $B_0=tfrac{1}{2}(B_1+B_3)$ and $B_5=tfrac{1}{2}(B_2+B_4)$; these points lie on the line $ell(x,y)=0$. Finally, let $B_infty$ be the point at infinity on this line. Let $mathfrak E$ be the pencil of conics with the base points $B_1,B_2,B_3,B_4$. Then the composition of the $B_infty$-switch and of the $B_0$-switch on the pencil $mathfrak E$ is the Kahan discretization of a Hamiltonian vector field $f=ell(x,y)begin{pmatrix}partial H/partial y -partial H/partial x end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map $Phi_f:mathbb C P^2dashrightarrowmathbb C P^2$ has three singular points $B_0,B_2,B_4$, while the inverse map $Phi_f^{-1}$ has three singular points $B_1,B_3,B_5$.
We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order $O(epsilon^2)$ in the coefficients of the discretization, where $epsilon$ is the stepsize.
Kahan discretization is applicable to any system of ordinary differential equations on $mathbb R^n$ with a quadratic vector field, $dot{x}=f(x)=Q(x)+Bx+c$, and produces a birational map $xmapsto widetilde{x}$ according to the formula $(widetilde{x}-x)/epsilon=Q(x,widetilde{x})+B(x+widetilde{x})/2+c$, where $Q(x,widetilde{x})$ is the symmetric bilinear form corresponding to the quadratic form $Q(x)$. When applied to integrable systems, Kahan discretization preserves integrability much more frequently than one would expect a priori, however not always. We show that in some cases where the original recipe fails to preserve integrability, one can adjust coefficients of the Kahan discretization to ensure its integrability.
Coisotropic deformations of algebraic varieties are defined as those for which an ideal of the deformed variety is a Poisson ideal. It is shown that coisotropic deformations of sets of intersection points of plane quadrics, cubics and space algebraic curves are governed, in particular, by the dKP, WDVV, dVN, d2DTL equations and other integrable hydrodynamical type systems. Particular attention is paid to the study of two- and three-dimensional deformations of elliptic curves. Problem of an appropriate choice of Poisson structure is discussed.
We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in $mathbb{P}^{n+2}$ satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space $W$ of dimension $n+2$, classify $n$-tuples of skew-symmetric 2-forms $A^{alpha} in Lambda^2(W)$ such that [ phi_{beta gamma}A^{beta}wedge A^{gamma}=0, ] for some non-degenerate symmetric $phi$.
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