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Darboux-covariant differential-difference operators and dressing chains

Sergey Leble
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Abstract

The general approach to chain equations derivation for the function generated by a Miura transformation analog is developing to account evolution (second Lax equation) and illustrated for Sturm-Liouville differential and difference operators. Polynomial differential operators case is investigated. Covariant sets of potentials are introduced by a periodic chain closure. The symmetry of the system of equation with respect to permutations of the potentials is used for the direct construction of solutions of the chain equations. A "time" evolution associated with some Lax pair is incorporated in the approach via closed t-chains. Both chains are combined in equations of a hydrodynamic type. The approach is next developed to general Zakharov-Shabat differential and difference equations, the example of 2x2 matrix case and NS equation is traced.

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Sergey Leble. 2005-04-01. Darboux-covariant differential-difference operators and dressing chains. https://arxiv.org/abs/math-ph/0504003

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