Harry Dym Hierarchy Research Articles

Virasoro symmetries of the constrained dispersionless mKP hierarchy

This paper aims at the auto-Bäcklund transformations for the dispersionless mKP hierarchy, dispersionless Miura map between the constrained dispersionless mKP hierarchy and the constrained dispersionless Harry Dym hierarchy and Virasoro symmetries of the constrained dispersionless mKP hierarchy. The additional symmetries of the constrained dispersionless mKP hierarchy are constructed in a manner similar to that of the dispersionless mKP hierarchy. By comparison, a new Laurent series Yl is introduced. The additional flows form a subalgebra of the Virasoro algebra.

Geometry of Inhomogeneous Poisson Brackets, Multicomponent Harry Dym Hierarchies, and Multicomponent Hunter–Saxton Equations

We introduce a natural class of multicomponent local Poisson structures $\mathcal P_k + \mathcal P_1$, where $\mathcal P_1$ is a local Poisson bracket of order one and $\mathcal P_k$ is a homogeneous Poisson bracket of odd order $k$ under assumption that is has Darboux coordinates (Darboux-Poisson bracket) and non-degenerate. For such brackets we obtain the general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples) and provide the description of the Casimirs, using purely algebraic procedure. In two-component case we completely classify such brackets up to the point transformation. From the description of Casimirs we derive new Harry Dym (HD) hierarchies and new Hunter-Saxton (HS) equations for arbitrary number of components. In two component case our HS equation differs from the well-known HS2 equation.

Reciprocal transformations of Harry–Dym hierarchy

In this paper, we work out the reciprocal transformations generated by the adjoint eigenfunctions, and write down the corresponding changes of the Lax operators, eigenfunctions and adjoint eigenfunctions under this kind of new reciprocal transformations at first. Then we discuss the successive actions of the reciprocal transformations generated by the eigenfunctions and the adjoint eigenfunctions. Also we investigate the commutativity of different types of the reciprocal transformations. The above results are very useful to construct the new type solutions.

Neumann Type Integrable Reduction to the Negative-Order Coupled Harry–Dym Hierarchy

Based on the Lax compatibility, the negative-order coupled Harry--Dym (ncHD) hierarchy depending upon one parameter $\alpha$ is retrieved in the Lenard scheme, which includes the two-component Camassa--Holm (2CH) equation as a special member with $\alpha=-\frac14$. By using a symmetric constraint, it is found that only in the case of $\alpha>1$ the ncHD hierarchy can be reduced to a family of backward Neumann type systems by separating the temporal and spatial variables on the tangent bundle of a unit sphere. The resultant backward Neumann type systems are proved to be completely integrable in the Liouville sense via a Lax equation. Finally, for $\alpha>1$, the relation between the ncHD hierarchy and the backward Neumann type systems is established, where the involutive solutions of backward Neumann type systems yield the finite parametric solutions to the ncHD hierarchy.

Algebro-Geometric Solutions of the Harry Dym Hierarchy

AbstractThe Harry Dym hierarchy is derived with the help of Lenard recursion equations and zero curvature equation. Based on the Lax matrix, an algebraic curve$\mathcal{K}_{n}$of arithmetic genus$n$is introduced, from which the corresponding meromorphic function$\phi$and Dubrovin-type equations are given. Further, the divisor and asymptotic properties of$\phi$are studied. Finally, algebro-geometric solutions for the entire hierarchy are obtained according to above results and the theory of algebraic curve.

Two kinds of finite-dimensional integrable reduction to the Harry–Dym hierarchy

In this paper, two kinds of finite-dimensional integrable reduction are studied for the Harry–Dym (HD) hierarchy. From the nonlinearization of Lax pair, the HD hierarchy is reduced to a class of finite-dimensional Hamiltonian systems (FDHSs) in view of a Bargmann map and a set of Neumann type systems by a Neumann map, which separate temporal and spatial variables on the symplectic space [Formula: see text] and the tangent bundle of ellipsoid [Formula: see text], respectively. It turns out that involutive solutions of the resulted finite-dimensional integrable systems (FDISs) directly give rise to finite parametric solutions of HD hierarchy through the Bargmann and Neumann maps. The finite-gap potential to the high-order stationary HD equation is obtained that cuts out a finite-dimensional invariant subspace for the HD flows. Finally, some comparisons of two kinds of integrable reductions are then discussed.

New extension of dispersionless Harry Dym hierarchy

The symmetry constraint for dispersionless Harry Dym (dHD) hierarchy is derived for the first time by taking dispersionless limit of that for 2+1 dimensional Harry Dym hierarchy. Then, the dHD is extended by means of the symmetry constraint which we derived. From the zero-curvature equation of the new extended dHD hierarchy, two types of dHD equations with self-consistent sources (dHDESCS) together with their associated conservation equations are obtained. Moreover, the hodograph solutions to the first type of dHDESCS are given. Finally, Bäcklund transformation between the extended dispersionless mKP hierarchy and extended dHD Hierarchy are also constructed.

Relation between the Negative-Order Harry Dym Hierarchy and a Family of Backward Neumann Type Systems

From the bidirectional Lenard gradients, the negative-order Harry Dym (nHD) hierarchy is retrieved and further embedded into a bi-Hamiltonian structure displaying integrability. It follows from Neumann type integrable reduction that the nHD hierarchy is reduced to a family of backward Neumann type systems, which separate the temporal and spatial variables on the tangent bundle of an ellipsoid. Backward Neumann type systems are then proved to be completely integrable in the Liouville sense. From the commutativity of backward Neumann type flows, the relation between the nHD hierarchy and backward Neumann type systems is specified, where the involutive solutions of backward Neumann type systems yield the finite parametric solutions of the nHD hierarchy. Moreover, we propose the concept of a negative-order Novikov equation that cuts out a finite-dimensional invariant subspace for a negative-order integrable system, which paves an alternative way to obtain explicit solutions of negative-order integrable nonlin...

Harry-Dym Hierarchy: Hereditary Operator and Ablowitz-Ramani-Segur Conjecture

Abstract. Using Lax representation, hereditary operator related to the Harry-Dym ( HD ) hierarchy is constructed. Also Lie analysis and Lax pair are used to drive the conserved quantities for the HD hierarchy. Lie symmetry method is also applied to study Painlev´e test for the family of HD hierarchy. It is found that for this family the Painlev´e property can be at most sufficient for integrability, but not necessary. This fact has been proved earlier for just first member of this family.

Links between(γn,σk)-KP Hierarchy,(γn,σk)-mKP Hierarchy, and (2+1)-(γn,σk)-Harry Dym Hierarchy

The new (2+1)-(γn,σk)-Harry Dym hierarchy and(γn,σk)-mKP hierarchy with two new time seriesγnandσk, which consist ofγn-flow,σk-flow, and mixedγnandσkevolution equations of eigenfunctions, are proposed. Gauge transformations and reciprocal transformations between(γn,σk)-KP hierarchy,(γn,σk)-mKP hierarchy, and (2+1)-(γn,σk)-Harry Dym hierarchy are studied. Their soliton solutions are presented.

Invertible Coupled KdV and Coupled Harry Dym Hierarchies

In this paper, we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one‐forms, recursion operators, Poisson and symplectic operators. We show that the invertible coupled KdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order.

A non-standard Lax formulation of the Harry Dym hierarchy and its supersymmetric extension

For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from that of the Korteweg–de Vries (KdV) equation through a reciprocal transformation. By supersymmetrizing this Lax operator, a new N = 2 supersymmetric extension of the Harry Dym hierarchy is constructed, and is further shown to be linked to one of the N = 2 supersymmetric KdV equations through the superconformal transformation. The bosonic limit of this new N = 2 supersymmetric Harry Dym equation is related to a coupled system of KdV–MKdV equations.

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.

A General Formula of Flow Equations for Harry—Dym Hierarchy

In this paper, we derive a general formula of the flow equtions for the Harry—Dym hierarchy. And three applications in n-reduction, (2 + 1)-dimensional generalization, and Kupershmidt reduction, are considered.

Construction of coupled Harry Dym hierarchy and its solutions from Stäckel systems

In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD) hierarchy from classical Stäckel separable systems. Both nonlocal and purely differential parts of hierarchies are obtained. We also construct various classes of solutions of cHD hierarchy from solutions of corresponding Stäckel systems.

An extended Harry Dym hierarchy

An extended Harry Dym hierarchy is constructed by using eigenfunctions and adjoint eigenfunctions of the spectral problems of the Harry Dym hierarchy associated with the pseudo-differential operator L = u∂ + u0 + u1∂−1 + ⋅⋅⋅. The corresponding Lax presentation possesses a self-consistent source involving squared eigenfunctions. The resulting extended Harry Dym hierarchy is reduced to the Harry Dym hierarchy with self-consistent sources under the n-reduction, Ln = (Ln)⩾2, and the k-constrained Harry Dym hierarchy under the k-constraint, Lk = (Lk)⩾2 + ∑Ni = 1qi∂−1ri∂2. A few particular examples are computed, together with their Lax pairs.

Multi-component Harry–Dym hierarchy and its integrable couplings as well as their Hamiltonian structures

This paper obtains the multi-component Harry–Dym (H–D) hierarchy and its integrable couplings by using two kinds of vector loop algebras G̃3 and G̃6. The Hamiltonian structures of the above system are given by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.

Bi-Hamiltonian aspects of a matrix Harry Dym hierarchy

We study the Harry Dym hierarchy of nonlinear evolution equations from the bi-Hamiltonian view point. This is done by using the concept of an S-hierarchy. It allows us to define a matrix Harry Dym hierarchy of commuting Hamiltonian flows in two fields that projects onto the scalar Harry Dym hierarchy. We also show that the conserved densities of the matrix Harry Dym equation can be found by means of a Riccati-type equation.

Additional symmetries and Bäcklund transformations for the dispersionless Harry Dym hierarchy

We give a dispersionless Toda-like extension to the dispersionless Harry Dym (dDym) hierarchy. The extended dDym (EdDym) hierarchy has a dressing formulation, and its underlying solution structure can be investigated through the twistor construction. We show that additional symmetries of the solution space generate Backlund transformations of the EdDym hierarchy. We present some examples of constructing new solutions of the (2+1)-dimensional dDym and EdDym equations via Backlund transformations.

Hydrodynamic reductions of the dispersionless Harry Dym hierarchy

We investigate the reductions of dispersionless Harry Dym hierarchy to systems of finitely many partial differential equations. These equations must satisfy the compatibility condition and they are diagonalizable and semi-Hamiltonian. By imposing a further constraint, the compatibility is reduced to a system of algebraic equations, whose solutions are described.