Analytic solutions and integrability for bilinear recurrences of order six

Some integrable maps and their Hirota bilinear forms

“…and then apply the total difference operator Δ to both sides; from this it follows that β defined as above is a first integral for (19). For all N, it can also be checked that (19) has the first integralζ…”

“…Sequences generated by certain bilinear recurrences of Somos type also admit further bilinear relations of higher order (see [33], for example), with the coefficients being first integrals, and this has been used to obtain first integrals for four-term Somos-6 and Somos-7 recurrences, in [19] and [8], respectively. Another approach to finding first integrals, based on reduction of conservation laws for the discrete KP or BKP equations, was used in [28].…”

“…The other results in section 3 are based on the connection between bilinear equations and cluster algebras, as explained in [8], which leads to a Poisson structure for the lifted DTKQ equation (19). For N even, both bilinear equations (5) and (6) reveal the connection with reductions of Hirota's discrete KdV equation; while for N odd, the second bilinear equation (7) leads to a link with reductions of a discrete time Toda equation, as well as an associated Bäcklund transformation (or BT, in the sense of [25]).…”

“…The complete proof of theorem 1.1, for arbitrary N, is reserved until section 3, where we begin by deriving (5) via a modified version of (2) (see equation (22) below), before treating the rest of the result and its detailed consequences for even/odd N separately. The interpretation in terms of Liouville integrability is naturally achieved by considering a lift of (2) to dimension N + 1, obtained by eliminating the parameter β: this yields equation (19) in section 3. A schematic picture of the connections between the main equations involved, valid for arbitrary N, is provided by the following diagram:…”

“…Thus we can apply the method described in [19], starting with particular numerical values of the initial conditions and looking for the smallest integer q such that a matrix M of size q 2 + 1, with entries M ij = τ q+i−j τ i+j−2 , has vanishing determinant: this corresponds to a bilinear relation of minimal order, with constant coefficients. By considering matrix entries of the form M ij = τ q+ i−j+k τ i+j+k−2 , for different choices of offset k, one can also obtain minimal order relations whose coefficients have period > 1.…”