Systems of conservation laws with third-order Hamiltonian structures

Abstract

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 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 α ∈ Λ 2 (W ) such that φ βγ A β ∧ A γ = 0, for some non-degenerate symmetric φ.MSC: 37K05, 37K10, 37K20, 37K25. 1

WDVV equations and invariant bi-Hamiltonian formalism

“…The above results imply that in the case N = 3 the WDVV quasilinear first-order systems are linearly degenerate, non diagonalizable and in the Temple class, as it follows from the main results in [35]. Indeed, the presence of third-order operators yields many interesting properties of the underlying first-order quasilinear system of PDEs.…”

“…The line congruence is linear if there are n linear relations between the Plücker coordinates; these are n linear line complexes. As it was proved in [35], every first-order quasilinear system of PDEs admitting a third-order HHO is associated with a linear line congruence.…”

“…There are interesting implications of the conjecture. Indeed, it was proved in [33,35] that third-order HHOs can be regarded as distinguished projective varieties, namely, quadratic line complexes. They determine families of varieties, linear line congruences, that correspond to first-order quasilinear systems of PDEs (first-order WDVV being one such systems in all known cases).…”

Bi-Hamiltonian structure of the oriented associativity equation

“…Remark 2 A direct computation shows that the two flows V i and w j αk u k xx commute if and only if the conditions (23e) and (23f) hold true. Moreover, by arguments that are similar to those of [12,Theorem 1] it can be proved that (23d) is a consequence of the other equations (23) and (12), (14).…”

Computing with Hamiltonian operators

“…The operator (36) can be rewritten in flat coordinates of −g js d is k as in (31). This means that the operator is completely determined by its leading term (g ij ) using (33). The pseudo-Riemannian metric (g ij ) is in bijective correspondence with certain projective varieties, see [31,32,33].…”