bi-Hamiltonian Formulation Research Articles

A combined Kaup–Newell type integrable hierarchy with four potentials and its bi-Hamiltonian formulation

This paper aims to introduce a Kaup–Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct its associated Liouville integrable Hamiltonian hierarchy, through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is shown by determining its recursion operator and bi-Hamiltonian formulation. An illustrative example of combined derivative nonlinear Schrödinger equations with two arbitrary constants is explicitly presented.

A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problem

This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation. A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out, which exhibits the Liouville integrability of each model in the resulting hierarchy. Two specific examples, consisting of novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations, are given.

Four-component combined integrable equations possessing bi-Hamiltonian formulations

This paper aims to generate a Liouville integrable Hamiltonian hierarchy by introducing a specific matrix eigenvalue problem with four components. The adopted approach is the zero curvature formulation. A bi-Hamiltonian formulation is furnished through applying the trace identity, which shows the Liouville integrability of the resulting hierarchy. Two examples of generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations are presented.

A Generalized Hierarchy of Combined Integrable Bi-Hamiltonian Equations from a Specific Fourth-Order Matrix Spectral Problem

The aim of this paper is to analyze a specific fourth-order matrix spectral problem involving four potentials and two free nonzero parameters and construct an associated integrable hierarchy of bi-Hamiltonian equations within the zero curvature formulation. A hereditary recursion operator is explicitly computed, and the corresponding bi-Hamiltonian formulation is established by the so-called trace identity, showing the Liouville integrability of the obtained hierarchy. Two illustrative examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations with four components and two adjustable parameters.

Four-Component Liouville Integrable Models and Their Bi-Hamiltonian Formulations

We aim at presenting Liouville integrable Hamiltonian models with four dependent variables from a specific matrix eigenvalue problem. The Liouville integrability of the resulting models is exhibited through formulating their bi-Hamiltonian formulations. The basic tools are the Lax pair approach and the trace identity. Two illustrative examples consist of novel four-component coupled integrable models of second-order and third-order

A four-component hierarchy of combined integrable equations with bi-Hamiltonian formulations

Upon introducing a specific 4 × 4 matrix eigenvalue problem with four components, we would like to construct a Liouville integrable Hamiltonian hierarchy, within the zero curvature formulation. Bi-Hamiltonian formulations are furnished via the trace identity, through which the Liouville integrability of the resulting hierarchy is explored. The first two nonlinear examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations.

On a fermionic extension of [formula omitted] equation

A fermionic extension of K(−1,−2) equation, alias a super K(−1,−2) equation, proposed by Tempesta et al. (2003), is converted to the N=1 supersymmetric Korteweg–de Vries equation via invertible transformations involving both independent and dependent variables. As implementation of this intimate connection, some integrable properties are established for the fermionic extension of K(−1,−2) equation, including a linear spectral problem in terms of 3 × 3 matrices, a Bäcklund transformation, a nonlinear superposition formula, infinitely many conservation laws, as well as its bi-Hamiltonian formulation.

Multicomponent Fokas–Lenells equations on Hermitian symmetric spaces

Multi-component integrable generalizations of the Fokas–Lenells equation, associated with each irreducible Hermitian symmetric space are formulated. Description of the underlying structures associated to the integrability, such as the Lax representation and the bi-Hamiltonian formulation of the equations is provided. Two reductions are considered as well, one of which leads to a nonlocal integrable model. Examples with Hermitian symmetric spaces of all classical series of types A.III, BD.I, C.I and D.III are presented in details, as well as possibilities for further reductions in a general form.

On the quest for generalized Hamiltonian descriptions of 3D-flows generated by the curl of a vector potential

We examine Hamiltonian analysis of three-dimensional advection flow [Formula: see text] of incompressible nature [Formula: see text] assuming that the dynamics is generated by the curl of a vector potential [Formula: see text]. More concretely, we elaborate Nambu–Hamiltonian and bi-Hamiltonian characters of such systems under the light of vanishing or non-vanishing of the quantity [Formula: see text]. We present an example (satisfying [Formula: see text]) which can be written as in the form of Nambu–Hamiltonian and bi-Hamiltonian formulations. We present another example (satisfying [Formula: see text]) which we cannot able to write it in the form of a Nambu–Hamiltonian or bi-Hamiltonian system while it can be manifested in terms of Hamiltonian one-form and yields generalized or vector Hamiltonian equations [Formula: see text].

On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations

Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schr\"odinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system which admits several conserved quantities, which allow to construct explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.

Two integrable couplings of a generalized D-Kaup–Newell hierarchy and their Hamiltonian and bi-Hamiltonian structures

We enlarge the spectral problem of a generalized D-Kaup–Newell (D-KN) spectral problem. Solving the enlarged zero-curvature equations, we produce integrable couplings. A reduction of the spectral matrix leads to a second integrable coupling system. Next, bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra are computed to employ the variational identity. The variational identity is then applied to the original enlarged spectral problem of a generalized D-KN hierarchy and the reduced problem. Hamiltonian structures are presented, as well as a bi-Hamiltonian formulation of the reduced problem. Both hierarchies have infinitely many commuting high-order symmetries and conserved densities, i.e., are Liouville integrable.

Poisson Brackets after Jacobi and Plücker

We construct a symplectic realization and a bi-hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Pl\" ucker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.

Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation

Abstract In this paper we study a (2+1)-dimensional coupling system with the Korteweg-de Vries equation, which is associated with non-semisimple matrix Lie algebras. Its Lax-pair and bi-Hamiltonian formulation were obtained and presented in the literature. We utilize Lie symmetry analysis along with the (G′/G)–expansion method to obtain travelling wave solutions of this system. Furthermore, conservation laws are constructed using the multiplier method.

Dissipative prolongations of the multipeakon solutions to the Camassa–Holm equation

Multipeakons are special solutions to the Camassa–Holm equation. They are described by an integrable geodesic flow on a Riemannian manifold. We present a bi-Hamiltonian formulation of the system explicitly and write down formulae for the associated first integrals. Then we exploit the first integrals and present a novel approach to the problem of the dissipative prolongations of multipeakons after the collision time. We prove that an n-peakon after a collision becomes an n−1-peakon for which the momentum is preserved.

Integrable [formula omitted]-invariant peakon equations from the NLS hierarchy

Two integrable U(1)-invariant peakon equations are derived from the NLS hierarchy through the tri-Hamiltonian splitting method. A Lax pair, a recursion operator, a bi-Hamiltonian formulation, and a hierarchy of symmetries and conservation laws are obtained for both peakon equations. These equations are also shown to arise as potential flows in the NLS hierarchy by applying the NLS recursion operator to flows generated by space translations and U(1)-phase rotations on a potential variable. Solutions for both equations are derived using a peakon ansatz combined with an oscillatory temporal phase. This yields the first known example of a peakon breather. Spatially periodic counterparts of these solutions are also obtained.

Generalized negative flows in hierarchies of integrable evolution equations

A one-parameter generalization of the hierarchy of negative flows is introduced for integrable hierarchies of evolution equations, which yields a wider (new) class of non-evolutionary integrable nonlinear wave equations. As main results, several integrability properties of these generalized negative flow equation are established, including their symmetry structure, conservation laws, and bi-Hamiltonian formulation. (The results also apply to the hierarchy of ordinary negative flows). The first generalized negative flow equation is worked out explicitly for each of the following integrable equations: Burgers, Korteweg-de Vries, modified Korteweg-de Vries, Sawada-Kotera, Kaup-Kupershmidt, Kupershmidt.

Some new integrable systems of two-component fifth-order equations

In this work, we develop some fifth-order integrable coupled systems of weight 0 and 1 which possess seventh-order symmetry. We establish four new systems, where in some cases, related recursion operator and bi-Hamiltonian formulations are given. We also investigate the integrability of the developed systems.

An integrable generalization of the super AKNS hierarchy and its bi-Hamiltonian formulation

Based on a Lie super-algebra B(0, 1), an integrable generalization of the super AKNS iso-spectral problem is introduced and its corresponding generalized super AKNS hierarchy is generated. By making use of the super-trace identity (or the super variational identity), the resulting super soliton hierarchy can be put into a super bi-Hamiltonian form. A generalized super AKNS soliton hierarchy with self-consistent sources is also presented.

An integrable extension of a soliton hierarchy associated with so(3,ℝ) and its bi-Hamiltonian formulation

A hierarchy of soliton equations is constructed from an extended spectral problem associated with the three-dimensional special orthogonal real Lie algebra so(3,ℝ) by means of zero curvature equations. The bi-Hamiltonian structure of this hierarchy is also presented via the trace identity.

On bi-Hamiltonian formulation of the perturbed Kepler problem

The perturbed Kepler problem is shown to be a bi-Hamiltonian system in spite of the fact that the graph of the Hamilton function is not a hypersurface of translation, which goes against a necessary condition for the existence of the bi-Hamiltonian structure according to the Fernandes theorem. In fact, the initial and perturbed Kepler systems are both isochronous systems and, therefore, the Fernandes theorem cannot be applied to them.