On the integrability of stationary and restricted flows of the KdV hierarchy

Abstract

Abstract. A bi-Hamiltonian formulation for stationary flows of the KdV hierarchy is derived in an extended phase space. A map between stationary flows and restricted flows is constructed: in a case it connects an integrable Hénon-Heiles system and the Garnier system. Moreover a new integrability scheme for Hamiltonian systems is proposed, holding in the standard phase space.

Quasi–periodic and periodic solutions for coupled nonlinear Schrödinger equations of Manakov type

“…The integrability of this case and separability in ellipsoidal coordinates was proved by Wojciechowski [48] (see also [27,44]). We employ this result to integrate the system in terms of ultraelliptic functions (hyperelliptic functions of the genus two curve) and then execute reduction of hyperelliptic functions to elliptic ones by imposing additional constrains on the parameters of the system.…”

Quasi-bi-Hamiltonian systems and separability

“…As a matter of fact, it is in general quite difficult to construct directly a biHamiltonian structure for a given integrable Hamiltonian vector field; so one can try to use some reduction procedure, starting from a few "universal" Poisson structures defined in an extended phase space. On the other hand, in the case of finite-dimensional systems arising as restricted or stationary flows from soliton equations [3,4], the final result of the reduction procedure are some physically interesting dynamical systems (for example the Hénon-Heiles system) which, in their natural phase space, satisfy a weaker condition than the biHamiltonian one. So, the notion of quasi-biHamiltonian (QBH) system can be introduced [5,6]; it was applied in [7] to dynamical systems with two degrees of freedom.…”

“…In this section we present two separable QBH systems with three and four degrees of freedom; they belong to a family of integrable flows obtained in [4] as stationary flows of the Korteweg-de Vries hierarchy [9]. This family contains the classical Hénon-Heiles system as its second member, so the higher members can be considered as multi-dimensional extensions of Hénon-Heiles.…”

Solving negative and mixed Toda hierarchies by Jacobi inversion problems

“…Just as the sl (2) Gaudin model corresponding to the N-gap sector of AKNS hierarchy [24] and the Garnier system corresponding to the N-gap sector of KdV hierarchy [25], the finite dimensional integrable system derived from V λ is closely related to the N-gap solution of cKdV hierarchy [26], Toda hierarchy [22], some (2 + 1)-Toda equation, and KP equation [27].…”