Positive and Negative Hierarchies of Integrable Lattice Models Associated with a Hamiltonian Pair

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Semi-direct sums of Lie algebras and continuous integrable couplings (2006, background): “ two above cases. Case of α = 1 and β = 0: Let us ﬁrst compute an example of the hierarchy (3.19). Assume that α = 1 and β = 0 for convenience, which corresponds to the lattice hierarchy presented in [26]. In this case, we have n0 = 0 in (2.14). It directly follows from (3.15) that ( f i = a (1) i v”
A modified Toda spectral problem and its hierarchy of bi-Hamiltonian lattice equations (2004, background): “24) which was previously presented in [12].”
Semidirect sums of Lie algebras and discrete integrable couplings (2006, background): “9) becomes the following lattice equation presented in [26]: rt1 = r(s (−1) − s) + r(r − r((1))), st1 = s(r − r ((1))).”