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Covariance of Lax pairs and integrability of the compatibility condition. (English. Russian original) Zbl 0989.35117
Theor. Math. Phys. 128, No. 1, 890-905 (2001); translation from Teor. Mat. Fiz. 128, No. 1, 65-83 (2001).
Summary: We study the joint covariance of Lax pairs (LPs) with respect to Darboux transformations (DTs). The scheme is based on comparing general expressions for the transformed coefficients of a LP and its Frechet derivative. We use the compact expressions of the DTs via a version of non-abelian Bell polynomials. We show that the so-called binary version of Bell polynomials forms a convenient basis for specifying the invariant subspaces. Some nonautonomous generalizations of KdV and Boussinesq equations are discussed in this context. We consider a Zakharov-Shabat-like problem to obtain restrictions at a minimal operator level. The subclasses that allow a DTs symmetry (covariance at the LP level) are considered from the standpoint of dressing-chain equations. The cases of the classical DT and binary combinations of elementary DTs are considered with possible reduction constraints of the Mikhailov type (generated by an automorphism). Examples of Liouville-von Neumann equations for the density matrix are considered as illustrations.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
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