Abstract
We study the integrability of a family of birational maps obtained as reductions of the discrete Hirota equation, which are related to travelling wave solutions of the lattice KdV equation. In particular, for reductions corresponding to waves moving with rational speed N/M on the lattice, where N, M are coprime integers, we prove the Liouville integrability of the maps when N + M is odd, and prove various properties of the general case. There are two main ingredients to our construction: the cluster algebra associated with each of the Hirota bilinear equations, which provides invariant (pre)symplectic and Poisson structures; and the connection of the monodromy matrices of the dressing chain with those of the KdV travelling wave reductions.
Highlights
The discrete Hirota equation [29] is an integrable bilinear partial difference equation for a function T = T(n1, n2, n3) of three independent variables, namelyTn1+1Tn1−1 = Tn2+1Tn2−1 + Tn3+1Tn3−1, where for brevity we take Tn1±1 = T(n1 ± 1, n2, n3), and for shifts in the n2 and n3 directions
We focus on the discrete KdV family, that is plane wave reductions of the discrete Hirota equation that take the form τm+2M+N τm = a τm+2M τm+N + b τm+M+N τm+M, M, N ∈ N, (2)
It would be interesting to use the above spectral coordinates on the hyperelliptic curves (28) to derive explicit formulae for the solutions of the iterated maps corresponding to the lattice KdV travelling wave reductions, as has been done for solutions of the discrete potential KdV equation in [28]
Summary
These kinds of recurrences inherit a Lax representation from the Lax representation of the discrete Hirota equation [14] Their non-autonomous versions are associated with q-Painleve equations and their higher order analogues [12, 21, 22], and they appear in the context of supersymmetric gauge theories and dimer models [1,2,3, 9]. In recent work [13, 14], two families of discrete Hirota reductions (1) associated with twodimensional lattice equations of discrete KdV/discrete Toda type have been studied. Both of them admit Lax representations which generate first integrals. We extend the results of [11], where the particular family of (N, 1) travelling waves was considered, and give complete proofs of various assertions made concerning the case of general reductions of type (N, M) in [14]
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