tri-Hamiltonian Structure Research Articles

Tri-Hamiltonian structure of the Ablowitz–Ladik hierarchy

We construct a local tri-Hamiltonian structure of the Ablowitz–Ladik hierarchy, and compute the central invariants of the associated bihamiltonian structures. We show that the central invariants of one of the bihamiltonian structures are equal to 124, and the dispersionless limit of this bihamiltonian structure coincides with the one that is defined on the jet space of the Frobenius manifold associated with the Gromov–Witten invariants of local P1. This result provides support for the validity of Brini’s conjecture on the relation of these Gromov–Witten invariants with the Ablowitz–Ladik hierarchy.

A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

Bäcklund transformations for tri-Hamiltonian dual structures of multi-component integrable systems

Bäcklund transformations for tri-Hamiltonian dual structures of multi-component integrable systems

Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański

We present first heavenly equation of Pleba\'nski in a two-component evolutionary form and obtain Lagrangian and Hamiltonian representations of this system. We study all point symmetries of the two-component system and, using the inverse Noether theorem in the Hamiltonian form, obtain all the integrals of motion corresponding to each variational (Noether) symmetry. We derive two linearly independent recursion operators for symmetries of this system related by a discrete symmetry of both the two-component system and its symmetry condition. Acting by these operators on the first Hamiltonian operator $J_0$ we obtain second and third Hamiltonian operators. However, we were not able to find Hamiltonian densities corresponding to the latter two operators. Therefore, we construct two recursion operators, which are either even or odd, respectively, under the above-mentioned discrete symmetry. Acting with them on $J_0$, we generate another two Hamiltonian operators $J_+$ and $J_-$ and find the corresponding Hamiltonian densities, thus obtaining second and third Hamiltonian representations for the first heavenly equation in a two-component form. Using P. Olver's theory of the functional multi-vectors, we check that the linear combination of $J_0$, $J_+$ and $J_-$ with arbitrary constant coefficients satisfies Jacobi identities. Since their skew symmetry is obvious, these three operators are compatible Hamiltonian operators and hence we obtain a tri-Hamiltonian representation of the first heavenly equation. Our well-founded conjecture applied here is that P. Olver's method works fine for nonlocal operators and our proof of the Jacobi identities and bi-Hamiltonian structures crucially depends on the validity of this conjecture.

Integrable cluster dynamics of directed networks and pentagram maps

The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatibility of Poisson and cluster structures and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. The appendix relates the simplest of these discrete maps to the Toda lattice and its tri-Hamiltonian structure.

Superintegrable cases of four-dimensional dynamical systems

Degenerate tri-Hamiltonian structures of Shivamoggi and generalized Raychaudhuri equations are exhibited. For certain specific values of the parameters, it is shown that hyperchaotic L\"{u} and Qi systems are superintegrable and admit tri-Hamiltonian structures.

4-Dimensional Frobenius manifolds and Painleve’ VI

A Frobenius manifold has tri-Hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We study the case of the lowest nontrivial dimension $$n=4$$ and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the Painleve $$\hbox {VI}\mu $$ equation. Since the solutions of this equation are known to parametrize semisimple Frobenius manifolds of dimension $$n=3$$ , this leads to an explicit procedure mapping 3-dimensional Frobenius structures of 4-dimensional ones, and giving all tri-Hamiltonian structures in four dimensions. We illustrate the construction by computing two examples in the framework of Frobenius structures on Hurwitz spaces.

A HIERARCHY OF SOLITON EQUATIONS ASSOCIATED WITH A HIGHER-DIMENSIONAL LOOP ALGEBRA AND ITS TRI-HAMILTONIAN STRUCTURE

A new higher-dimensional loop algebra is given for which a Lax isospectral problem is set up whose compatibility condition gives rise to a Liouville integrable soliton hierarchy along with eight-component potential functions. Specially, the hierarchy of evolution equations has a tri-Hamiltonian structure obtained by the trace identity.

A new loop algebra [formula omitted] and its applications to integrable system

We construct a new loop algebra A ∼ 5 which is used to set up a linear isospectral problem. By making use of Tu scheme, a new integrable couplings of the generalized MKdV hierarchy is obtained. Then its tri-Hamiltonian structure is constructed by the quadratic-form identity. Furthermore, we point out that the integrable couplings is integrable in the sense of Liouville.

A generalized MKdV hierarchy, tri-Hamiltonian structure, higher-order binary constrained flows and its integrable couplings system

A subalgebra of higher-dimension loop algebra A 2 ˜ is constructed, from which a new 3 × 3 isospectral problem is designed. By making use of Tu’s scheme, an integrable Hamiltonian hierarchy of equations in the sense of Liouville is obtained, which possesses tri-Hamiltonian structure. As reduction cases of the hierarchy presented in this paper, the generalized MKdV equation is engendered. By establishing binary symmetric constraints, three constrained flows of the hierarchy are presented, which are then reduced to Hamiltonian systems. Finally, an integrable coupling system is obtained by constructing a high-dimension loop algebra.

The Hamiltonian Structures of the Two-Dimensional Toda Lattice and R-Matrices

We construct the tri-Hamiltonian structure of the two-dimensional Toda hierarchy using the R-matrix theory.

A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations

A subalgebra of loop algebra A 2 is first constructed, which has its own special feature. It follows that a new Liouville integrable hierarchy of evolution equations is obtained, possessing a tri-Hamiltonian structure, which is proved by us in this paper. Especially, three symplectic operators are constructed directly from recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. As reduction cases of the hierarchy presented in this paper, the celebrated MKdV equation and heat-conduction equation are engendered, respectively. Therefore, we call the hierarchy a generalized MKdV-H system. At last, a high-dimension loop algebra G is constructed by making use of a proper scalar transformation. As a result, a type expanding integrable model of the MKdV-H system is given.

Tri-Hamiltonian Toda Lattice and a Canonical Bracket for Closed Discrete Curves

Flows on (or variations of) discrete curves in \(\mathbb{R}^2 \) give rise to flows on a subalgebra of functions on that curve. For a special choice of flows and a certain subalgebra this is described by the Toda lattice hierarchy. Here it is shown that the canonical symplectic structure on \(\mathbb{R}^{2N} \), which can be interpreted as the phase space of closed discrete curves in \(\mathbb{R}^2 \) with length N, induces Poisson commutation relations on the above-mentioned subalgebra which yield the tri-Hamiltonian poisson structure of the Toda lattice hierarchy.

Tri-Hamiltonian Structures of Egorov Systems of Hydrodynamic Type ∗

We prove a simple condition under which the metric corresponding to a diagonalizable semi-Hamiltonian hydrodynamic type system belongs to the class of Egorov (potential) metrics. For Egorov diagonal hydrodynamic type systems satisfying natural semisimplicity and homogeneity conditions, we prove necessary and sufficient conditions under which the third structure is local or corresponds to a metric of constant curvature. The results are illustrated by some well-known physical examples of such systems.

A coupled AKNS–Kaup–Newell soliton hierarchy

A coupled AKNS–Kaup–Newell hierarchy of systems of soliton equations is proposed in terms of hereditary symmetry operators resulted from Hamiltonian pairs. Zero curvature representations and tri-Hamiltonian structures are established for all coupled AKNS–Kaup–Newell systems in the hierarchy. Therefore all systems have infinitely many commuting symmetries and conservation laws. Two reductions of the systems lead to the AKNS hierarchy and the Kaup–Newell hierarchy, and thus those two soliton hierarchies also possess tri-Hamiltonian structures.

Similarity Reductions and Painlevé Properties for the Kupershmidt Equations

Five types of similarity reductions of the Kupershmidt equations which admit a tri-Hamiltonian structure are found by a direct method. Two types of reduction equations which are Painlevé II and IV types are coincident with those obtained by classical Lie approach. Both algebraic and logarithmic branch points for time t can be entered into the solutions of Kupershmidt equations. The integrability of the Kupershmidt equations is re-examined by the singularity analysis using the Weiss-Kruskal approach and the Ablowitz-Ramani-Segur algorithm.

On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure

The multi-Hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed. Infinitely many conservation laws had earlier been obtained for these equations. Starting from a (primary) Hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi-Hamiltonian structure are obtained and it is shown that the second Hamiltonian operator can be constructed solely through a knowledge of the first Hamiltonian function. The recursion operator which first appears at the level of bi-Hamiltonian structure gives rise to an infinite sequence of conserved Hamiltonians. It is found that in general there exist two different infinite sequences of conserved quantities for these equations. The recursion relation defining higher Hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the (k+1)st Hamiltonian operator which depends on the kth Hamiltonian function. The infinite sequence of conserved Hamiltonians are common to all the higher Hamiltonian structures. The equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri-Hamiltonian structure with two distinct infinite sets of conserved quantities. The isothermal case of γ=1 is an exceptional one that requires separate treatment. This corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to Poisson’s equation in nonlinear acoustics.

Canonical structures for dispersive waves in shallow water

The canonical Hamiltonian structure of the equations of fluid dynamics obtained in the Boussinesq approximation are considered. New variational formulations of these equations are proposed and it is found that, as in the case of the KdV equation and the equations governing long waves in shallow water, they are degenerate Lagrangian systems. Therefore, in order to cast these equations into canonical form it is again necessary to use Dirac’s theory of constraints. It is found that there are primary and secondary constraints which are second class and it is possible to construct the Hamiltonian in terms of canonical variables. Among the examples of Boussinesq equations that are discussed are the equations of Whitham–Broer–Kaup which Kupershmidt has recently expressed in symmetric form and shown to admit tri-Hamiltonian structure.

Reply to comment of R. L. Glick

The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatibility of Poisson and cluster structures and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. The appendix relates the simplest of these discrete maps to the Toda lattice and its tri-Hamiltonian structure.