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Weakly nonlocal Hamiltonian structures: Lie derivative and compatibility. (English) Zbl 1138.37043

Under certain technical assumptions, all Hamiltonian structures compatible with a given nondegenerate Hamiltonian structure \(P\) are the Lie derivatives of \(P\) along suitable nonlocal vector fields. The author extends this idea to the infinite-dimensional Hamiltonian structures. In particular Hamiltonian structures of arbitrary order (including e.g. operators of Dubrovin-Novikov type) and compatible with a given Hamiltonian structure of zeroth- or first-order are complete described.
In more detail, let \(A\) be the algebra of smooth functions of varibbles \(x,u_0,u_1,\dots\), where \(u_k= (u^1_k,\dots, u^n_k)\in\mathbb{R}^n\) are vectors, let \(\text{Mat}_q(A)[[D^{-1}]]\) be the space of formal series \[ L= \sum^{j=k}_{j=-\infty} h_j D^j\quad (D=\partial/\partial x+ \sum u_{j+1}\partial/\partial u_j), \] where \(h_j\) are \(q\times q\)-matrices. This \(L\) is called weakly nonlocal if \[ L_-= \sum^{j=-1}_{j=-\infty} h_j D^j= \sum^{\alpha= k}_{\alpha= 1} f_\alpha\otimes D^{-1}\circ g_\alpha \] for appropriate \(f_\alpha,g_\alpha\in A^q\) and \(L\) is called Hamiltonian if the Schouten square vanishes: \([L,L]= 0\). The author search for these Hamiltonian operators compatible with a given weakly nonlocal operator \(P\) by picking a weakly nonlinear \(Q\) and requiring the Lie derivative \(L_Q(P)\) to be Hamiltonian.

MSC:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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