Integrable lattices (Q1568004)

In this paper integrable lattices corresponding to the Hamiltonians \[ H= p_xq_x+ V(p,q_x, q)\tag{1} \] are constructed, where \(V\) is a quadratic polynomial in \(p\) and \(q_x\). The authors are interested in only the two cases \[ H= p_xq_x+ q^2_x(\varepsilon p^2+\alpha p+ \beta)+ p^2(\gamma q_x+ \delta),\tag{2} \] where \(\varepsilon\), \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are arbitrary parameters, and \[ H= p_x q_x- q^2_x p^2- p^2 r(q)- \textstyle{{1\over 2}} pr'(q)- \textstyle{{1\over 12}} r''(q),\tag{3} \] where \(r(q)\) is an arbitrary fourth-order polynomial in \(q\). Different choices of the parameters in (2) correspond to numerous integrable generalizations of the nonlinear Schrödinger equation. The authors discuss the Toda, Volterra and Heisenberg models in detail and obtain totally discrete Lagrangians. Moreover, they discuss the relation of this systems to the Hirota equations.