Walter Oevel

Constrained KP hierarchy and bi-Hamiltonian structures

The Kakomtsev-Petviashvili (KP) hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions. Constraining the KP flows in terms of squared eigenfunctions one obtains 1+1-dimensional integrable equations with scattering problems given by pseudo-differential Lax operators. The bi-Hamiltonian nature of these systems is shown by a systematic construction of two general Poisson brackets on the algebra of associated Lax-operators. Gauge transformations provide Miura links to modified equations. These systems are constrained flows of the modified KP hierarchy, for which again a general description of their bi-Hamiltonian nature is given. The gauge transformations are shown to be Poisson maps relating the bi-Hamiltonian structures of the constrained KP hierarchy and the modified KP hierarchy. The simplest realization of this scheme yields the AKNS hierarchy and its Miura link to the Kaup-Broer hierarchy.

The Bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems

The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.

Darboux theorems and Wronskian formulas for integrable systems: I. Constrained KP flows

Abstract Generalizations of the classical Darboux theorem are established for pseudo-differential scattering operators of the form L = ∑ i=0 N u i ∂ i + Σ i=1 m Φ i ∂ −1 Ψ i † i. Iteration of the Darboux transformations leads to a gauge transformed operator with coefficients given by Wronskian formulas involving a set of eigenfunctions of L. Nonlinear integrable partial differential equations are associated with the scattering operator L which arise as a symmetry reduction of the multicomponent KP hierarchy. With a suitable linear time evolution for the eigenfunctions the Darboux transformation is used to obtain solutions of the integrable equations in terms of Wronskian determinants.

A 3 sx 3 matrix spectral problem for AKNS hierarchy and its binary nonlinearization

A 3 × 3 matrix spectral problem for AKNS soliton hierarchy is introduced and the corresponding Bargmann symmetry constraint involving Lax pairs and adjoint Lax pairs is discussed. An explicit new Poisson algebra is proposed and thus the Liouville integrability is established for the nonlinearized spatial system ind a hierarchy of nonlinearized temporal systems under the control of the nonlinearized spatial system. The obtained nonlinearized Lax systems, in which the nonlinearized spatial system is intimately related to stationary AKNS flows, lead to a sort of new involutive solutions to each AKNS soliton equation. Therefore, the binary nonlinearization theory is successfully extended to a case of 3 × 3 matrix spectral problem for AKNS hierarchy.

Explicit formulas for symmetries and conservation laws of the Kadomtsev-Petviashvili equation

Abstract Two nonlocal recursion operators are given, which yield explicit formulas for infinite hierarchies of symmetry generators and conservation laws for the two-dimensional Korteweg-de Vries equation. It is shown that the constants of the motion are in involution and that the symmetries commute.

GAUGE TRANSFORMATIONS AND RECIPROCAL LINKS IN 2 + 1 DIMENSIONS

Generalized Lax equations are considered in the spirit of Sato theory. Three decompositions of an underlying algebra of pseudo-differential operators lead, in turn, to three different classes of integrable nonlinear hierarchies. These are associated with Kadomtsev-Petviashvili, modified Kadomtsev-Petviashvili and Dym hierarchies in 2 + 1 dimensions. Miura- and auto-Backlund transformations are shown to originate naturally from gauge transformations of the Lax operators. General statements on reciprocal links between these hierarchies are established, which, in particular, give rise to novel reciprocal auto-Backlund transformations for the Dym hierarchy. These links are formulated as Darboux theorems for the associated Lax operators.

Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure

Basic invariants, such as conserved quantities, symmetries, mastersymmetries, and recursion operators are explicitly constructed for the following nonlinear lattice systems: The modified Korteweg–de Vries lattice, the Ablowitz–Ladik lattice, the Brusci–Ragnisco lattice, the Ragnisco–Tu lattice and some cases of the class of integrable systems introduced by Bogoyavlensky. The algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the Lax pair analysis is discussed. By explicit presentation of the higher‐order members of the corresponding hierarchies new completely integrable lattice flows are found. For all systems, multi‐Hamiltonian formulations are given.

R-Matrices and Higher Poisson Brackets for Integrable Systems

The tri-hamiltonian nature of Lax-equations is revealed: starting with an R-matrix on an associative algebra g equipped with a trace form there are g compatible Poisson brackets with linear, quadratic and cubic dependence on the coordinates. The invariant functions (Casimir functions) on g* are in involution relative to these brackets, they yield a hierarchy of integrable tri-hamiltonian Lax-equations. The results can be applied to solvable PDE’s such as the Korteweg-de Vries equation as well as to finite integrable systems such as the Toda lattice. In these cases the Poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well-known recursion operators.

The bi‐Hamiltonian structure of some nonlinear fifth‐ and seventh‐order differential equations and recursion formulas for their symmetries and conserved covariants

Using a bi‐Hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth‐ and seventh‐order nonlinear partial differential equations; among them, the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and the Kupershmidt equation. We show that the Lie algebras of the symmetry groups of these equations are of a very special form: Among the C∞ vector fields they are generated from two given commuting vector fields by a recursive application of a single operator. Furthermore, for some higher order equations, those multisoliton solutions, which for ‖t‖→∞ asymptotically decompose into traveling wave solutions, are characterized as eigenvector decompositions of certain operators.

MASTERSYMMETRIES, ANGLE VARIABLES, AND RECURSION OPERATOR OF THE RELATIVISTIC TODA LATTICE

Conserved quantities, bi‐Hamiltonian formulation, and recursive structure of the relativistic Toda lattice (RT) are obtained in an algorithmic way without making use of the Lax representation. Furthermore, for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries. A new hierarchy of completely integrable systems is discovered, which turns out to correspond to the ‘‘negative’’ of the hierarchy of RT. Thus it is shown that the full algebra of time‐dependent symmetry group generators for each member of the RT hierarchy is isomorphic to the algebra of first order differential operators with Laurent polynomials as coefficients. The surprising phenomenon is revealed that the members of the RT hierarchy are connected to their negative counterparts by explicit Backlund transformation.