Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

Abstract

The integrability of an m-component system of hydrodynamic type, u t = V (u)u x , by the generalized hodograph method requires the diagonalizability of the m × m matrix V (u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains -infinite-component systems of hydrodynamic type for which the ∞×∞ matrix V (u) is 'sufficiently sparse'. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.MSC: 35L40, 35L65, 37K10.

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

“…It turns out that all these three different nonlinear partial differential equations possess the same infinite set of N-component hydrodynamic reductions parameterised by N arbitrary functions of a single variable [18,19] (we note that the solutions to these N-component reductions are parameterised, in their turn, by another N arbitrary functions of a single variable). This property was used in [12,13](see also [48], [14], [20], [34]) when introducing the integrability criterion for a wide class of kinetic equations, corresponding hydrodynamic chains and 2+1 quasilinear equations. Moreover, it was proved in [36] that the existence of at least one N-component hydrodynamic reduction written in the so-called symmetric form is sufficient for integrability in the sense of [12].…”

Differential geometry of hydrodynamic Vlasov equations

“…With these hypotheses, the system can then be integrated by the generalized hodograph transformation ( [3]). We remark that the semiHamiltonian property is automatically satisfied for a Hamiltonian system with Dubrovin-Novikov Hamiltonian structure, and that the conditions for the system to be respectively diagonalizable, or semi-Hamiltonian, can be written invariantly; each corresponds to the vanishing of some tensor ( [4], [1]). In particular, for the diagonalizability condition, if one defines the Nijenhuis tensor of the matrix v i j by: and then the Haantjes tensor by:…”

“…where A = (A 0 , A 1 , ...) t is an infinite column vector and V (A) is an ∞ × ∞ matrix, with the following properties (see [1], [6]), 1) for every row only finitely many elements are nonzero 2) every element of the matrix depends only on a finite number of variables.…”

Classification of integrable hydrodynamic chains and generating functions of conservation laws

“…in both above equations one can obtain (2) and the infinite series of conservation laws for ∂A m = 0, m > k + 1 (8) and corresponding 2+1 quasilinear equations (see, for instance, (4)) was found in [10], [11], [9], [12], [19], [23], [32]). Recently some of these integrable hydrodynamic chains were rediscovered (see [2]) and studied in [2], [4], [5], [22], [25], [24]).…”

“…The existence of an extra conservation law (an external method), successfully (an incomplete classification) used for the Kupershmidt Poisson bracket and for the (an incomplete classification) Kupershmidt-Manin Poisson bracket (see [19]); 3. Vanishing of the Haantjes tensor (an internal method), successfully (unfinished classification) used for the integrable hydrodynamic chains (8) and successfully (complete classification) for the Kupershmidt-Manin Poisson bracket (see [9]);…”