Harry Dym Hierarchy Research Articles

A completely integrable system associated with the Harry-Dym hierarchy

By use of nonlinearization method about spectral problem, a classical completely integrable system associated with the Harry-Dym (HD) hierarchy is obtained. Furthermore, the involutive solution of each equation in the HD hierarchy is presented, in particular, the involutive solution of the well-known HD equation ut = (u- 1 2 )xxx is given.

Multicomponent Schwarzian KdV hierarchies

We extend recent results of Dorfman [1] and Wilson [2] on the Schwarzian-KdV equation to the coupled KdV and Harry Dym hierarchies isospectral to an energy dependent Schrödinger operator [3, 4]. Whilst the N-component coupled KdV hierarchies posses ( N + 1) differential Hamiltonian structures, their Schwarzian modifications posses ( N + 1) nonlocal Hamiltonian structures whose inverses are purely differential. We present, as examples, the Schwarzian modifications associated with the dispersive water waves (DWW) and the Harry Dym equations.

The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane.

The Korteweg-de Vries, modified Korteweg-de Vries, and Harry Dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane. These purely local dynamics conserve an infinite number of global geometric properties of the curves, such as perimeter and enclosed area. Several techniques used to study these integrable systems are shown to have simple differential-geometric interpretations. Connections with incompressible, inviscid fluid flow in two dimensions are suggested

The coupled HD hierarchy and a classical integrable system of the complex form

The coupled Harry Dym (HD) equations are discussed by means of the complex form of the real involutive system. Using the nonlinearization of Lax pairs of the coupled HD equation, a finite-dimensional completely integrable system in the Liouville sense is obtained. By making use of the solutions of commutative flows, the representation of the solutions for the hierarchy of the HD equation are generated.

Finite-dimensional completely integrable systems associated with the Harry Dym and the coupled Harry Dym hierarchies

Under the Bargmann constraints between the potentials and the eigenfunctions, the spectral problems q xx + λvq= αq and q xx + λuq + λ 2 vq= αq respectively associated with the Harry Dym and the coupled Harry Dym equations are nonlinearized to be completely integrable systems in the Liouville sense.

Coupled Harry Dym equations with multi-Hamiltonian structures

The authors consider the isospectral flows of ( delta 2+ Sigma 1N upsilon i lambda i) Psi = alpha Psi . Using an unusual form of the 'Lax approach' they derive in a particularly simple manner: (a) the 'second half' of the Lax pair; (b) the associated Hamiltonian structures; (c) an infinite hierarchy of Poisson commuting Hamiltonians. In this way they show that these equations possess (N+1) compatible, purely differential Hamiltonian structures. The case N=1 is just the bi-Hamiltonian Harry Dym hierarchy. They thus extend the recent results on multi-Hamiltonian coupled KdV equations.