Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations

Abstract

Abstract. We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive at a complete classification of irreducible solutions.

Hamiltonian systems of Jordan block type: Delta-functional reductions of the kinetic equation for soliton gas

“…In particular, if det(g ij ) = 0, then g = {g ij } is a contravariant flat metric and Γ ij k = −g is Γ j sk where Γ j sk are Christoffel symbols of the associated Levi-Civita connection. The matrix A = (A i j ) of the corresponding system (8) is given by the formula A i j = ∇ i ∇ j h. Thus, to specify a Hamiltonian structure of type (9), it is sufficient to provide the corresponding contravariant flat metric g.…”

“…Ansatz (2) with η i (x, t) = const was discussed in [10]. In this case, the last n equations (3) are satisfied identically, while the first n equations constitute an integrable diagonalisable linearly degenerate system whose Hamiltonian aspects (both local and nonlocal) were explored in [9,3]. The case of non-constant η i (x, t) was investigated recently in [16]: the matrix of the corresponding system (3) is reducible to n Jordan blocks of size 2 × 2, furthermore, it was shown that the system is linearly degenerate and integrable by a suitable extension of the generalised hodograph method of [34,35].…”