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Integrable lattices. (English. Russian original) Zbl 0984.37093
Theor. Math. Phys. 118, No. 2, 173-182 (1999); translation from Teor. Mat. Fiz. 118, No. 2, 217-228 (1999).
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.

##### MSC:
 37K60 Lattice dynamics; integrable lattice equations 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems 35Q55 NLS equations (nonlinear Schrödinger equations)
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