The modular vector field plays an important role in the theory of Poisson manifolds and is intimately connected with the Poisson cohomology of the space. In this paper we investigate its significance in the theory of integrable systems. We illustrate in detail the case of the Toda lattice both in Flaschka and natural coordinates.
We prove that the classical, non–periodic Toda lattice is super–integrable. In other words, we show that it possesses 2N − 1 independent constants of motion, where N is the number of degrees of freedom. The main ingredient of the proof is the use of some special action–angle coordinates introduced by Moser to solve the equations of motion.
We examine the multiple Hamiltonian structure and construct a symplectic realization of the Volterra model. We rediscover the hierarchy of invariants, Poisson brackets and master symmetries via the use of a recursion operator. The rational Volterra bracket is obtained using a negative recursion operator.
We consider an integrable system of reduced Maxwell–Bloch equations that describes the evolution of an electromagnetic field in a two-level medium that is inhomoge-neously broadened. We prove that the relevant Bäcklund transformation preserves the reality of the n-soliton potentials and establish their pole structure with respect to the broadening… (More)