Franco Magri

Reduction techniques for infinite dimensional Hamiltonian systems: some ideas and applications

In the language of tensor analysis on differentiable manifolds, we present a reduction method of integrability structures, and apply it to recover some well-known hierarchies of integrable nonlinear evolution equations.

Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV

We study a certain family of Schrödinger operators whose eigenfunctions ϕ(χ, λ) satisfy a differential equation in the spectral parameter λ of the formB(λ,∂λ)ϕ=Θ(x)ϕ. We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class ofbispectral potentials. This extends and complements a result of Duistermaat and Grünbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy.

Coisotropic Deformations of Associative Algebras and Dispersionless Integrable Hierarchies

The paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies, like the dispersionless KP and modified KP hierarchies and the universal Whitham hierarchy of genus zero. It stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras. It discusses the link between the structure constants and Hirota’s tau function, and shows that the dispersionless Hirota bilinear equations are, within this approach, a way of writing the associativity conditions for the structure constants in terms of the tau function. It also suggests a simple interpretation of the algebro-geometric construction of the universal Whitham equations of genus zero due to Krichever.

Bihamiltonian Geometry and Separation of Variables for Toda Lattices

Abstract We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical Hamilton–Jacobi method in the so-called Darboux–Nijenhuis coordinates.

Geometric Proof of Lie's Linearization Theorem

In 1883, S. Lie found the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general procedure for constructing linearizing transformations by using an over-determined system of four equations. We present here a simple geometric proof of the theorem, known as Lies linearization test, stating that the compatibility of Lies four auxiliary equations furnishes a necessary and sufficient condition for linearization.

Lax–Nijenhuis operators for integrable systems

The relationship between Lax and bi‐Hamiltonian formulations of dynamical systems on finite‐ or infinite‐dimensional phase spaces is investigated. The Lax–Nijenhuis equation is introduced and it is shown that every operator that satisfies that equation satisfies the Lenard recursion relations, while the converse holds for an operator with a simple spectrum. Explicit higher‐order Hamiltonian structures for the Toda system, a second Hamiltonian structure of the Euler equation for a rigid body in n‐dimensional space, and the quadratic Adler–Gelfand–Dickey structure for the KdV hierarchy are derived using the Lax–Nijenhuis equation.

Nijenhuis G-manifolds and Lenard bicomplexes: a new approach to KP systems

We suggest a method to extend the theory of recursion operators to integrable Hamiltonian systems in two-space dimensions, like KP systems. The approach aims to stress the conceptual unity of the theories in one and two space dimensions. A sound explanation of the appearance of bilocal operators is also given.

DISPERSIONLESS INTEGRABLE EQUATIONS AS COISOTROPIC DEFORMATIONS: EXTENSIONS AND REDUCTIONS

We discuss the interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain associative algebras and other algebraic structures. We show that with this approach, the dispersionless Hirota equations for the dKP hierarchy are just the associativity conditions in a certain parameterization. We consider several generalizations and demonstrate that B-type dispersionless integrable hierarchies, such as the dBKP and the dVN hierarchies, are coisotropic deformations of the Jordan triple systems. We show that stationary reductions of the dispersionless integrable equations are connected with dynamical systems on the plane that are completely integrable on a fixed energy level.

A note on fractional KdV hierarchies

One of the cornerstones of the theory of integrable systems of KdV type has been the remark that the n-GD (Gel’fand–Dickey) equations are reductions of the full Kadomtsev–Petviashvilij (KP) theory. In this paper we address the analogous problem for the fractional KdV theories, by suggesting candidates of the “KP theories” lying behind them. These equations are called “KP(m) hierarchies,” and are obtained as reductions of a bigger dynamical system, which we call the “central system.” The procedure allowing passage from the central system to the KP(m) equations, and then to the fractional KdVnm equations, is discussed in detail in the paper. The case of KdV32 is given as a paradigmatic example.

Bihamiltonian Manifolds And Sato’s Equations

This paper is a concise introduction to Sato’s equations from the point of view of Hamiltonian mechanics. It aims to show that the theory of soliton equations may be completely built on the study of the Casimir functions of a pencil of Poisson brackets on a Poisson manifold.