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Boundary conditions for two-dimensional integrable chains. (English) Zbl 1044.82517

Summary: The symmetry method of studying boundary value problems is generalized to the multi-dimensional case. In passing from \(1+1\) dimensions to \(2+1\) dimensions the main obstacle is the existence of nonlocal variables. To overcome this obstacle we have derived additional constraints that link the nonlocal variables of different levels. As an illustration, the application of the method to the two-dimensional Toda lattice is considered.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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References:

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