Bi-Hamiltonian structure in Frenet–Serret frame

New global solutions of the Jacobi partial differential equations

“…In particular, constant structure matrices also arise from Theorem 4 as D ψ -solutions. To see this, it suffices to consider the value ρ = n. In such case, there are (n − ρ) = 0 functions D l of the form (8) and J in (9) is entirely given by submatrix J [1] , in other words J = J [1] . Moreover, functions J [1] ij = ψ ij (D ρ+1 (x), .…”

“…where J [1] , J [2] , J [3] and J [4] are submatrices of sizes ρ × ρ, ρ × (n − ρ), (n − ρ) × ρ and (n−ρ)×(n−ρ), respectively. In the rest of the proof, the entries of J [k] will be denoted J…”

“…Proof. Submatrix J [1] is skew-symmetric by definition. Let us now prove that submatrices J [2] and J [3] also verify skew-symmetry.…”

“…This completes the proof of the theorem. ✷ Of course, in the solution family described in Theorem 4 the linear dependences among the elements of J have been chosen in such a way that the ρ first rows and columns (those conforming J [1] ) span the rest of rows and columns, as it is clear from equations (11)(12)(13)(14)(15). Actually, this choice is entirely arbitrary and was used without loss of generality.…”

“…Together, equations (1) and (2) are often referred to as the Jacobi PDEs. In expression (1) and in what follows, it is ∂ l ≡ ∂/∂x l . For the sake of clarity, the tensor index notation sometimes used in this context will be avoided, since its use does not provide any particular advantage for the analysis to come.…”