Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
1995-05-17
Nonlinear Sciences
Exactly Solvable and Integrable Systems
Scientific paper
10.1088/0305-4470/28/23/021
The problem of construction of the boundary conditions for the Toda lattice compatible with its higher symmetries is considered. It is demonstrated that this problem is reduced to finding of the differential constraints consistent with the ZS-AKNS hierarchy. A method of their construction is offered based on the B\"acklund transformations. It is shown that the generalized Toda lattices corresponding to the non-exceptional Lie algebras of finite growth can be obtained by imposing one of the four simplest integrable boundary conditions on the both ends of the lattice. This fact allows, in particular, to solve the problem of reduction of the series $A$ Toda lattices into the series $D$ ones. Deformations of the found boundary conditions are presented which leads to the Painlev\'e type equations. Key words: Toda lattice, boundary conditions, integrability, B\"acklund transformation, Lie algebras, Painlev\'e equations
Adler Vsevolod E.
Habibullin Ismagil T.
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