multi-Hamiltonian Structure Research Articles

Multi-Hamiltonian structure of the epidemics model accounting for vaccinations and a suitable test for the accuracy of its numerical solvers

We derive a generalized Hamiltonian formalism for a modified susceptible–infectious–recovered/removed (SIR) epidemic model taking into account the population V of vaccinated persons. The resulting SIRV model is shown to admit three possible functionally independent Hamiltonians and hence three associated Poisson structures. The reduced case of vanishing vaccinated sector shows a complete correspondence with the known Poisson structures of the SIR model. The SIRV model is shown to be expressible as an almost Nambu system, except for a scale factor function breaking the divergenceless property. In the autonomous case with time-independent stationary ratios k and b, the SIRV model is shown to be a maximally super-integrable system. For this case we test the accuracy of numerical schemes that are suited to solve the stiff set of SIRV differential equations.

General soliton solutions to a coupled Fokas–Lenells equation

In this paper, we firstly establish the multi-Hamiltonian structure and infinitely many conservation laws for the vector Kaup–Newell hierarchy of the positive and negative orders. The first nontrivial negative flow corresponds to a coupled Fokas–Lenells equation. By constructing a generalized Darboux transformation and using a limiting process, all kinds of one-soliton solutions are constructed including the bright–dark soliton, the dark–anti-dark soliton and the breather-like solutions. Furthermore, multi-bright and multi-dark soliton solutions are derived and their asymptotic behaviors are investigated.

Formula omitted] Toda systems

We define an integrable Hamiltonian system of Toda type associated with the real Lie algebra so(p,q). As usual there exist a periodic and a non-periodic version. We construct, using the root space, two Lax pair representations and the associated Poisson tensors. We prove Liouville integrability and examine the multi-Hamiltonian structure. The system is a projection of a canonical An type Toda lattice via a Flaschka transformation. It is also obtained via a complex change of variables from the classical Toda lattice.

Bi-Hamiltonian structure of an asymmetric heavenly equation

In the paper of Sheftel and Malykh (2009 J. Phys. A: Math. Theor. 42 395202) on the classification of second-order PDEs with four independent variables that possess partner symmetries, an asymmetric heavenly equation appears as one of the canonical equations admitting partner symmetries. Here, for the asymmetric heavenly equation formulated in a two-component form, we present the Lax pair of Olver–Ibragimov–Shabat type and obtain its multi-Hamiltonian structure. Therefore, by Magri’s theorem, it is a completely integrable bi-Hamiltonian system in four dimensions.

Soliton solutions by Darboux transformation and some reductions for a new Hamiltonian lattice hierarchy

In this paper, we start from the discrete spectral problem and construct a lattice hierarchy by properly choosing an auxiliary spectral problem , which can reduce to the Volterra hierarchy, the Ablowitz–Ladik hierarchy, positive and negative lattice hierarchies and a new hierarchy. The new hierarchy is integrable in involutory Lax's sense and possesses multi-Hamiltonian structure. In addition, the Darboux transformation of the lattice hierarchy is obtained when the freely adjustable function and m=1. Then some soliton solutions are obtained by using Darboux transformation. This method is also suitable for other more general spectral problems in mathematics and physics.

Self-dual gravity is completely integrable

We discover a multi-Hamiltonian structure of a complex Monge–Ampère equation (CMA) set in a real first-order 2-component form. Therefore, by Magri's theorem this is a completely integrable system in four real dimensions. We start with Lagrangian and Hamiltonian densities and obtain a symplectic form and the Hamiltonian operator that determines the Dirac bracket. We have calculated all point symmetries of the 2-component CMA system and Hamiltonians of the symmetry flows. We have found two new real recursion operators for symmetries which commute with the operator of a symmetry condition on solutions of the CMA system. These operators form two Lax pairs for the 2-component system. The recursion operators, applied to the first Hamiltonian operator, generate infinitely many real Hamiltonian structures. We show how to construct an infinite hierarchy of higher commuting flows together with the corresponding infinite chain of their Hamiltonians.

A lattice hierarchy and its continuous limits

By introducing a discrete spectral problem, we derive a lattice hierarchy which is integrable in Liouville's sense and possesses a multi-Hamiltonian structure. It is show that the discrete spectral problem converges to the well-known AKNS spectral problem under a certain continuous limit. In particular, we construct a sequence of equations in the lattice hierarchy which approximates the AKNS hierarchy as a continuous limit.

A Lattice Hierarchy with a Free Function and Its Reductions to the Ablowitz-Ladik and Volterra Hierarchies

By embedding a free function into a compatible zero curvature equation, we propose a lattice hierarchy with the free function which still admits zero curvature representation. It is interesting that the hierarchy can reduce the Ablowitz-Ladik hierarchy, the Volterra hierarchy and a new hierarchy by properly choosing the embedded function. Moreover, the new hierarchy is integrable in Liouville’s sense and possess multi-Hamiltonian structure.

A generalized Ablowitz–Ladik hierarchy, multi-Hamiltonian structure and Darboux transformation

Starting from a discrete spectral problem, we derive a hierarchy of nonlinear discrete equations which includes the Ablowitz–Ladik hierarchy and a new hierarchy as special cases. Especially, we investigate in detail the integrability and resolvability of the new hierarchy. It is shown that the new hierarchy is integrable in Liouville’s sense and possesses multi-Hamiltonian structure. A Darboux transformation is established for a typical discrete system in the new hierarchy with the help of the gauge transformation of its Lax pair. As applications of the Darboux transformation, new exact solutions of the discrete system are explicitly given.

The Poisson bracket compatible with the classical reflection equation algebra

We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the $XXX$ Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.

A family of the Poisson brackets compatible with the Sklyanin bracket

We introduce a family of compatible Poisson brackets on the space of 2 × 2 polynomial matrices, which contains the Sklyanin bracket, and use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet, the open and periodic Toda lattices, the discrete self-trapping model and the Goryachev–Chaplygin gyrostat.

Symmetries, integrals, and three-dimensional reductions of Plebañski’s second heavenly equation

We study symmetries and conservation laws for Plebanski’s second heavenly equation written as a first-order nonlinear evolutionary system which admits a multi-Hamiltonian structure. We construct an optimal system of one-dimensional subalgebras and all inequivalent three-dimensional symmetry reductions of the original four-dimensional system. We consider these two-component evolutionary systems in three dimensions as natural candidates for integrable systems.

Dispersionless noncommutative Kadomtsev–Petviashvili hierarchy: Multi-Hamiltonian structure and large N-like V-algebras

We propose a dispersionless limit of the NC Kadomtsev–Petviashvili hierarchy. Multi-Hamiltonian formulation, l-reductions and relevant W-algebras are investigated. As examples, Gelfand–Dickey (GD) Poisson brackets for dispersionless NCKdV and NC Boussinesq hierarchies are constructed explicitly. The associated GD Poisson algebras are shown large N- like analogs of classical V-algebras.

Multi-Hamiltonian structure of Plebanski's second heavenly equation

We show that Plebanski's second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magri's theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions.

On the bi-Hamiltonian structure of Bogoyavlensky–Toda lattices

In this paper we prove that the Bogoyavlensky–Toda lattices corresponding to simple, classical Lie groups are bi-Hamiltonian. A review of previous results on the multi-Hamiltonian structure of these systems is also included. The areas investigated include master symmetries, recursion operators, invariants and higher Poisson brackets. The case of exceptional, simple Lie groups is still an open problem.

Tri-Hamiltonian formulation for certain integrable lattice equations

A systematic investigation of integrable differential–difference equations with two independent variables admitting multi-Hamiltonian structure is presented. Considering the Volterra (VL), Toda (TL), Relativistic Toda (RT), Belov–Chaltikian (BC) and Blaszak–Marciniak both three (BM3) and four (BM4) coupled lattice equations it is shown that they admit a sequence of operators out of which only three are Hamiltonian ones and so they are tri-Hamiltonian systems only. It is observed that the constructed third operator for VL and BC lattice equations is Hamiltonian only if the field variable is periodic with even period.

Multi-Lagrangians for integrable systems

We propose a general scheme to construct multiple Lagrangians for completely integrable nonlinear evolution equations that admit multi-Hamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a conserved quantity higher/lower than the Hamiltonian in the potential part of the new Lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map. This leads to some remarkable new developments. We show that nonlinear evolutionary systems that admit N-fold first order local Hamiltonian structure can be cast into variational form with 2N−1 Lagrangians which will be local functionals of Clebsch potentials. This number increases to 3N−2 when the Miura transformation is invertible. Furthermore we construct a new Lagrangian for polytropic gas dynamics in 1+1 dimensions which is a free, local functional of the physical field variables, namely density and velocity, thus dispensing with the necessity of introducing Clebsch potentials entirely. This is a consequence of bi-Hamiltonian structure with a compatible pair of first and third order Hamiltonian operators derived from Sheftel’s recursion operator.

A Liouville integrable Hamiltonian system associated with a generalized Kaup–Newell spectral problem

Starting from a generalized Kaup–Newell spectral problem involving an arbitrary function, we derive a hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as Kaup–Newell equation, Chen–Lee–Liu equation, Gerdjikov–Ivanov equation, Burgers equation, modified Korteweg–de Vries equation and Sharma–Tasso–Olever equation. It is also shown that the hierarchy is integrable in Liouville's sense and possesses multi-Hamiltonian structure.

A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations

By introducing a spectral problem with an arbitrary parameter, we derive a Kaup–Newell-type hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as the Kundu equation, the Kaup–Newell (KN) equation, the Chen–Lee–Liu (CLL) equation, the Gerdjikov–Ivanov (GI) equation, the Burgers equation, the modified Korteweg-deVries (MKdV) equation and the Sharma–Tasso–Olver equation. It is shown that the hierarchy is integrable in Liouville’s sense and possesses multi-Hamiltonian structure. Under the Bargann constraint between the potentials and the eigenfunctions, the spectral problem is nonlinearized as a finite-dimensional completely integrable Hamiltonian system. The involutive representation of the solutions for the Kaup–Newell-type hierarchy is also presented. In addition, an N-fold Darboux transformation of the Kundu equation is constructed with the help of its Lax pairs and a reduction technique. According to the Darboux transformation, the solutions of the Kundu equation is reduced to solving a linear algebraic system and two first-order ordinary differential equations. It is found that the KN, CLL, and GI equations can be described by a Kundu-type derivative nonlinear Schrödinger equation involving a parameter. And then, we can construct the Hamiltonian formulations, Lax pairs and N-fold Darboux transformations for the Kundu, KN, CLL, and GI equations in explicit and unified ways.

Integrable systems of derivative nonlinear Schrödinger type and their multi-Hamiltonian structure

A spectral problem and the associated hierarchy of Schrodinger type equations are proposed. It is shown that the hierarchy is integrable in Liouville's sense and possesses multi-Hamiltonian structure. It is found that several kinds of important equation such as the Kaup-Newell (KN) equation, the Chen-Lee-Liu (CLL) equation, the Gerdjikov-Ivanov (GI) equation, the modified Korteweg-de Vries equation and the Sharma-Tasso-Olever equation are members in the hierarchy as its special reductions. Moreover, KN, CLL and GI equations are described by using a unified generalized derivative Schrodinger equation involving a parameter, and their Hamiltonian structure and Lax pairs are also given by unified and explicit formulae.