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.)

Zbl 0703.00014