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Register a new userSystems of conservation laws with third-order Hamiltonian structures
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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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Generalizing results by Bryant and Griffiths [Duke Math. J., 1995, V.78, 531-676], we completely describe local conservation laws of second-order (1+1)-dimensional evolution equations up to contact equivalence. The possible dimensions of spaces of conservation laws prove to be 0, 1, 2 and infinity. The canonical forms of equations with respect to contact equivalence are found for all nonzero dimensions of spaces of conservation laws.
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 $B_1,B_3,B_5$.
Let $V$ be a vector space of dimension $n+1$. We demonstrate that $n$-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank $n$ in $S^2(Lambda^2V)$ that lie in the kernel of the natural map $S^2(Lambda^2V)to Lambda^4V$. Non-equivalent operators correspond to different orbits of the natural action of $SL(n+1)$. Based on this result, we obtain a classification of such operators for $nleq 4$.
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.
We develop a theory of integrable dispersive deformations of 2+1 dimensional Hamiltonian systems of hydrodynamic type following the scheme proposed by Dubrovin and his collaborators in 1+1 dimensions. Our results show that the multi-dimensional situation is far more rigid, and generic Hamiltonians are not deformable. As an illustration we discuss a particular class of two-component Hamiltonian systems, establishing the triviality of first order deformations and classifying Hamiltonians possessing nontrivial deformations of the second order.
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