Eight Lectures on Integrable Systems

The sine-Gordon model with integrable defects revisited

“…Compatibility of two "time" evolutions a priori does not guarantee the existence of a Poisson structure which would render these two evolutions Hamiltonian. Were such a Poisson structure built, it only guarantees the Poisson commutation of the two corresponding Hamiltonians but not the existence of higher conserved Poisson commuting Hamiltonians, unless a Magri-type algorithm [26,27] allows to build a recursion operator from the two Hamiltonians and the Poisson structure, and hence to deduce the hierarchy of Poisson structure dual to the postulated hierarchy of hamiltonians. To formulate in another language: Equation (4.2) can be interpreted indeed as a Bäcklund transformation (see e.g.…”

“…[25] regarding the problems related to non-local and/or skew symmetric r-matrices and [26,28] for the issue of finding the quadratic Poisson structure "derived" from a linear dynamical r-matrix structure. Within our restricted conditions the bulk monodromy operators then obey a wellestablished quadratic Poisson algebra [29].…”

A Three-component Extension of the Camassa–Holm Hierarchy

“…In this section we review some facts on the geometry of bi-Hamiltonian manifolds (see, e.g., [19,5]), and we recall from [18] that the bi-Hamiltonian structure of the (usual) CH hierarchy can be obtained by means of a reduction. Let (M, P 1 , P 2 ) be a bi-Hamiltonian manifold, i.e., a manifold M endowed with two compatible Poisson tensors P 1 and P 2 .…”

Согласованные скобки Ли - Пуассона на алгебрах Ли $e(3)$ и $so(4)$

“…будем называть для краткости биинтегрируемыми системами или обобщенными бигамильтоновыми системами. Для того чтобы интегралы движения находились в биинволюции (1.1), достаточно, чтобы соответствующие им векторные поля X Hi были бигамильтоновыми [1], [2]. В этом случае интегралы движения H i образуют цепочку Ленарда-Магри и удовлетворяют соотношениям P dH 0 = 0, X Hi = P dH i = P dH i−1 , P dH n = 0.…”