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Dirac structures and integrability of nonlinear evolution equations. (English) Zbl 0717.58026

Nonlinear evolution equations: integrability and spectral methods, Proc. Workshop, Como/Italy 1988, Proc. Nonlinear Sci., 425-431 (1990).
[For the entire collection see Zbl 0703.00014.]
Let A be a Lie algebra and \((\Omega,d)=\{\oplus \Omega^ i\), \(i\geq 0\), d: \(\Omega\) \({}^ q\to \Omega^{q+1}\}\) a complex of linear spaces with \(i_ a: \Omega \to \Omega\), \(i_ a\Omega^ q\subset \Omega^{q-1}\), such that \(i_ ai_ b+i_ bi_ a=0\) and \([i_ ad+di_ a,i_ b]=i_{[a,b]}.\) For this complex (\(\Omega\),d) over A a pairing \((a,\xi)=i_ a\xi \in \Omega^ 0\) is given for \(\xi \in \Omega^ 1\) and \(a\in A\) and the Lie derivative \(L_ a=i_ ad+di_ a\) with \(L_ ab=[a,b]\) can be extended to analogues of tensor fields. If \(L_ aS=0\), S is called a symmetry and S is conserved along a. Examples are the de Rham complex over a manifold X or the formal variational calculus of Gelfand-Dikij. Given \(A\oplus \Omega^ 1\) equipped with a canonical symmetric bilinear form \(<h_ 1\oplus \xi_ 1,h_ 2\oplus \xi_ 2>=(h_ 1,\xi_ 2)+(h_ 2,\xi_ 1)\) for \(L\subset A\oplus \Omega^ 1\) one determines \(L^{\perp}\) via \(<, >=0\). A Dirac structure on (A,\(\Omega\)) is a linear subspace \(L\subset A\oplus \Omega^ 1\) satisfying \((1)\quad L^{\perp}=L\) and \((2)\quad (L_{h_ 1}\xi_ 2,h_ 3)+(L_{h_ 2}\xi_ 3,h_ 1)+(L_{h_ 3}\xi_ 1,h_ 2)=0\) for all \(h_ i\oplus \xi_ i\in L\). Connections to Poisson and symplectic structures are evident and there is an associated Hamiltonian formalism. Two Dirac structures \(L,M\subset A\oplus \Omega^ 1\) form a (compatible) pair if the set \(A_{LM}=\{a_ 1\oplus a_ 2:\) there exists \(\xi \in \Omega^ 1\) with \(a_ 1\oplus \xi \in M\), \(a_ 2\oplus \xi \in L\}\subset A\oplus \Omega^ 1\) is a Nijenhuis relation (Nijenhuis operators and relations are important in questions of integrability theory). The main result here is a generalization of the Lenard scheme of integrability in the context of Dirac structures. Examples are given and suggestions for further research are indicated.
Reviewer: R.Carroll

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Citations:

Zbl 0703.00014