Nonlocal Hamiltonian Structures Research Articles

Galilei-invariant and energy-preserving extensions of Benjamin–Bona–Mahony-type equations

The classical Benjamin–Bona–Mahony equation (BBM equation) models unidirectional propagation of long gravity surface waves of small amplitude. Unlike many other water wave models, it lacks the Galilean invariance, which is an essential property of physical systems. It is shown that by an addition of a higher asymptotic order nonlinear term, this deficiency can be corrected, giving rise to a new Galilei invariant Benjamin–Bona–Mahony equation (iBBM equation). Moreover, further additional higher-order terms can be chosen in a way that the augmented model preserves the energy conservation property along with Hamiltonian and Lagrangian structures. The resulting equation is referred to as energy-preserving Benjamin–Bona–Mahony equation (eBBM).It is shown that both the classical BBM equation and the energy-preserving eBBM equations belong to a one-parameter (α) family that shares essentially the same local and nonlocal symmetries, conservation laws, Hamiltonian, and Lagrangian structures, with the BBM and eBBM equations corresponding to parameter values α=0 and α=1, respectively. Symmetry and conservation law classifications reveal a special case α=1/3, which is shown to correspond to a rescaled version of the celebrated integrable Camassa–Holm (CH) equation. Local symmetries and conservation laws are computed, and numerical solution behaviour is compared for the three BBM-type modes and the CH-equivalent eBBM1/3 model.

Quasilinear Systems of First Order PDEs with Nonlocal Hamiltonian Structures

In this paper we wonder whether a quasilinear system of PDEs of first order admits Hamiltonian formulation with local and nonlocal operators. By using the theory of differential coverings, we find differential-geometric conditions necessary to write a given system with one of the three Hamiltonian operators investigated.

Purely nonlocal Hamiltonian formalism for systems of hydrodynamic type

We study purely nonlocal Hamiltonian structures for systems of hydrodynamic type. In the case of a semi-Hamiltonian system, we show that such structures are related to quadratic expansions of the diagonal metrics naturally associated with the system.

Hamiltonian Structures of Reductions of the Benney System

We show how to construct the Hamiltonian structures of any reduction of the Benney chain (dispersionless KP). The construction follows the scheme suggested by Ferapontov, leading in general to nonlocal Hamiltonian structures. In some special cases these reduce to local structures. All the geometric objects which define the Poisson bracket, the metric, connection and Riemann curvature, are obtained explicitly, in terms of the n-parameter family of conformal maps associated with the reduction.

Reciprocal transformations and flat metrics on Hurwitz spaces

We consider hydrodynamic systems which possess a local Hamiltonian structure. To such a system, there are also associated an infinite number of nonlocal Hamiltonian structures. We give the necessary and sufficient conditions so that, after a nonlinear transformation of the independent variables, the reciprocal system still possesses a local Hamiltonian structure. We show that, under our hypotheses, bi-Hamiltonicity is preserved by the reciprocal transformation. Finally, we apply such results to the reciprocal systems of genus g Whitham–KdV modulation equations.

Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility

We show that under certain technical assumptions any weakly nonlocal Hamil- tonian structure compatible with a given nondegenerate weakly nonlocal symplectic struc- ture J can be written as the Lie derivative of J 1 along a suitably chosen nonlocal vector field. Moreover, we present a new description for local Hamiltonian structures of arbitrary order compatible with a given nondegenerate local Hamiltonian structure of zero or first order, including Hamiltonian operators of the Dubrovin-Novikov type.

A Geometric Study of the Dispersionless Boussinesq Type Equation

We discuss the dispersionless Boussinesq type equation, which is equivalent to the Benney–Lax equation, being a system of equations of hydrodynamical type. This equation was discussed in [4]. The results include: A description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws (cosymmetries). Highly interesting are the appearances of operators that send conservation laws and symmetries to each other but are neither Hamiltonian, nor symplectic. These operators give rise to a noncommutative infinite-dimensional algebra of recursion operators.

(Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to theN= 1 supersymmetric KdV equation

Using methods of Kersten et al (2004 J. Geom. Phys. 50 273–302) and Krasil'shchik and Kersten (2000 Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Dordrecht: Kluwer)), we accomplish an extensive study of the N = 1 supersymmetric Korteweg–de Vries (KdV) equation. The results include a description of local and nonlocal Hamiltonian and symplectic structures, five hierarchies of symmetries, the corresponding hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws. We stress that the main point of the paper is not just the results on super-KdV equation itself, but merely exposition of the efficiency of the geometrical approach and of the computational algorithms based on it.

Reciprocal transformations of Hamiltonian operators of hydrodynamic type: Nonlocal Hamiltonian formalism for linearly degenerate systems

Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differential-geometric properties. We apply our results to linearly degenerate semi-Hamiltonian systems in Riemann invariants, a typical example being Rti=(∑m=1nRm−Ri)Rxi, i=1,2,…,n. Since all such systems are linearizable by appropriate (generalized) reciprocal transformations, our formulas provide an infinity of mutually compatible nonlocal Hamiltonian structures, explicitly parametrized by n arbitrary functions of one variable.

Prolongation algebras and Hamiltonian operators for peakon equations

We consider a family of non-evolutionary partial differential equations, labelled by a singleparameter b,all of which admit multi-peakon solutions. For the two special integrablecases, namely the Camassa–Holm and Degasperis–Procesi equations(b = 2and 3), we explain how their spectral problems have reciprocal links to Lax pairsfor negative flows, in the Korteweg–de Vries and Kaup–Kupershmidt hierarchiesrespectively. An analogous construction is presented in the case of theSawada–Kotera hierarchy, leading to a new zero-curvature representation for theintegrable Vakhnenko equation. We show how the two special peakonequations are isolated via the Wahlquist–Estabrook prolongation algebramethod. Using the trivector technique of Olver, we provide a proof of theJacobi identity for the non-local Hamiltonian structures of the wholepeakon family. Within this class of Hamiltonian operators (also labelled byb),we present a uniqueness theorem which picks out the special casesb = 2, 3.

A Simple Family of Non-Local Poisson Brackets

A dispersionless integrable system with repeated eigenvalues is presented. For N ≥ 3 components the system has no local Hamiltonian structure. Infinitely many simple compatible non-local Hamiltonian structures are given, using a result of Ferapontov.

The averaging of nonlocal Hamiltonian structures in Whitham′s method

We consider the m‐phase Whitham′s averaging method and propose the procedure of “averaging” nonlocal Hamiltonian structures. The procedure is based on the existence of a sufficient number of local‐commuting integrals of the system and gives the Poisson bracket of Ferapontov type for Whitham′s system. The method can be considered as the generalization of the Dubrovin‐Novikov procedure for the local field‐theoretical brackets.

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.