Super-trace Identity Research Articles

The Bargmann symmetry constraint of the new super Dirac hierarchy

For the orthosymplectic Lie superalgebra osp(2,2), we construct a new super Dirac integrable hierarchy, which can be written as the super Hamiltonian structure with the aid of supertrace identity. Upon choosing N distinct spectral parameters, a Bargmann symmetry constraint is proposed for the super Dirac integrable hierarchy. By substituting the symmetry constraint into N copies finite-dimensional super systems, we find the constrained super systems, defined over the supermanifold R4N|4N, are completely integrable in the Liouville sense.

Multi-component super integrable Hamiltonian hierarchies

By constructing a new type of multi-component Lie superalgebra sl(2N,1), a method of generation of multi-component super integrable hierarchies is proposed. We discuss two applications and then obtain a coupled generalized super integrable Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy and a multi-component generalized super integrable AKNS hierarchy respectively. Then, the super Hamiltonian structures of the resulting super integrable hierarchies are deduced by means of supertrace identity.

The generalized super Kaup–Newell equation and its super bi-Hamiltonian structure

In this work, from the generalized Kaup–Newell (KN) equation spectral problem, the generalized super KN equation and its bi-Hamiltonian structure are constructed based on a Lie super-algebras and the super-trace identity. Besides, a generalized super KN equation with self-consistent sources is also discussed.

On Construction of a Super Hierarchy of the Wadati-Konno-Ichikawa Equation

In this paper, a super Wadati-Konno-Ichikawa (WKI) hierarchy associated with a 3×3 matrix spectral problem is derived with the help of the zero-curvature equation. We obtain the super bi-Hamiltonian structures by using of the super trace identity. Infinitely, many conserved laws of the super WKI equation are constructed by using spectral parameter expansions.

A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi‐Hamiltonian structures

For the orthosymplectic Lie superalgebra , we choose a set of basis matrices. A linear combination of those basis matrices presents a spatial spectral matrix. The compatible condition of the spatial part and the corresponding temporal parts of the spectral problem leads to a generalized super AKNS (GSAKNS) hierarchy. By making use of the supertrace identity, the obtained GSAKNS hierarchy can be written as the super bi‐Hamiltonian structures.

New super integrable hierarchies associated with osp(2|2) and spo(2|2) and their applications

Based on the Lie superalgebras osp(2|2) and spo(2|2), we generate some new super soliton hierarchies. These hierarchies can reduce to the AKNS hierarchy and the super-AKNS hierarchy. And by using the supertrace identity on Lie superalgebras, we construct the super-Hamiltonian structures of these new super soliton hierarchies. Then, as the applications of spo(2|2) ≅ osp(2|2) and spo(2|2) ≅ sl(1|2), we obtain the relations of different hierarchies associated with different Lie superalgebras.

Constructing variable coefficient nonlinear integrable coupling super AKNS hierarchy and its self‐consistent sources

Constructing variable coefficient super integrable equation hierarchy is an important problem in soliton theory. In this letter, new Lax pairs with some arbitrary functions are proposed and a variable coefficient integrable coupling of super AKNS hierarchy is generated. The super Hamiltonian structures of variable coefficient coupling equation hierarchy is derived with the aid of the super trace identity. Furthermore, the self‐consistent sources of variable coefficient super integrable coupling hierarchy are established. It is indicated that this method is an efficient way to construct the variable coefficient super integrable equation hierarchy.

A super Sasa–Satsuma hierarchy and bi-Hamiltonian structures

A super Sasa–Satsuma hierarchy associated with a 3 × 3 matrix spectral problem is proposed with the aid of the zero-curvature equation and Lenard recursion equations. A typical member in the hierarchy is the super Sasa–Satsuma equation. The super bi-Hamiltonian structures of the super Sasa–Satsuma hierarchy are constructed by utilizing the super trace identity. The infinite conservation laws of the super Sasa–Satsuma equation are presented by resorting to the spectral parameter expansion.

A Integrable Generalized Super-NLS-mKdV Hierarchy, Its Self-Consistent Sources, and Conservation Laws

A generalized super-NLS-mKdV hierarchy is proposed related to Lie superalgebra B(0,1); the resulting supersoliton hierarchy is put into super bi-Hamiltonian form with the aid of supertrace identity. Then, the super-NLS-mKdV hierarchy with self-consistent sources is set up. Finally, the infinitely many conservation laws of integrable super-NLS-mKdV hierarchy are presented.

Super integrable hierarchies associated with spl(2, 1) and Darboux transformations

Based on the Lie superalgebra spl(2, 1) and the TAH scheme, we generate (1+1)-dimensional and (2+1)-dimensional super soliton hierarchies associated with spl(2, 1). And by using the super-trace identity on Lie superalgebras, we construct the super-Hamiltonian structures of these new super soliton hierarchies. At last, we also obtain the Darboux transformations of the (1+1)-dimensional super soliton hierarchy.

A new integrable symplectic map by the binary nonlinearization to the super AKNS system

Based on the constructed new Lie super-algebra from OSP(2,2), the super bi-Hamiltonian structure of a new super AKNS hierarchy is obtained by making use of super-trace identity. For the new super AKNS system, an explicit symmetry constraint between the potentials and the eigenfunctions is proposed. Moreover, the super AKNS system is decomposed into two compatible finite-dimensional super integrable systems and the obtained super systems are proved to be finite-dimensional super integrable Hamiltonian systems in the super-symmetry manifold R 4 N ∣ 4 N .

The Binary Nonlinearization of the Super Integrable System and Its Self-Consistent Sources

Abstract The super integrable system and its super Hamiltonian structure are established based on a loop super Lie algebra and super-trace identity in this paper. Then the super integrable system with self-consistent sources and conservation laws of the super integrable system are constructed. Furthermore, an explicit Bargmann symmetry constraint and the binary nonlinearization of Lax pairs for the super integrable system are established. Under the symmetry constraint,the n $n$ -th flow of the super integrable system is decomposed into two super finite-dimensional integrable Hamilton systems over the supersymmetric manifold. The integrals of motion required for Liouville integrability are explicitly given.

Nonlinear Super Integrable Couplings of Super Yang Hierarchy and Its Super Hamiltonian Structures

Nonlinear super integrable couplings of the super Yang hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of Yang hierarchy were obtained.

A generalized super AKNS hierarchy associated with Lie superalgebra sl(2|1) and its super bi-Hamiltonian structure

Starting from a 3 × 3 matrix-valued spectral problem associated with a Lie superalgebra sl(2|1), a generalized super Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy is derived. The resulting super AKNS hierarchy has a super bi-Hamiltonian structure by the supertrace identity.

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.

Nonlinear Super Integrable Couplings of a Super Integrable Hierarchy

Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. And its super Hamiltonian structures were established by using super trace identity. As its reduction, special cases of this nonlinear super integrable coupling were obtained.

A new six-component super soliton hierarchy and its self-consistent sources and conservation laws**Project supported by the National Natural Science Foundation of China (Grant Nos. 11547175, 11271008 and 61072147), the First-class Discipline of University in Shanghai, China, and the Science and Technology Department of Henan Province, China (Grant No. 152300410230).

A new six-component super soliton hierarchy is obtained based on matrix Lie super algebras. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy. After that, the self-consistent sources of the new six-component super soliton hierarchy are presented. Furthermore, we establish the infinitely many conservation laws for the integrable super soliton hierarchy.

Generalized fractional supertrace identity for Hamiltonian structure of NLS–MKdV hierarchy with self-consistent sources

In the paper, based on the modified Riemann–Liouville fractional derivative and Tu scheme, the fractional super NLS–MKdV hierarchy is derived, especially the self-consistent sources term is considered. Meanwhile, the generalized fractional supertrace identity is proposed, which is a beneficial supplement to the existing literature on integrable system. As an application, the super Hamiltonian structure of fractional super NLS–MKdV hierarchy is obtained.

Nonlinear Superintegrable Couplings for Supercoupled KDV Hierarchy with Self-Consistent Sources

In this paper, based on matrix Lie superalgebras, a new isospectral super-Lax matrix is established. By employing the super-Ablowitz–Kaup–Neweu–Segur (AKNS) scheme nonlinear superintegrable couplings for the supercoupled KdV hierarchy are worked out. Supertrace identity is used to furnish super-Hamiltonian structures for the superintegrable hierarchy. Furthermore, we propose nonlinear superintegrable couplings of the resulting integrable hierarchy with self-consistent sources.

Some 2+1 Dimensional Super-Integrable Systems

Abstract In the article, we make use of the binormial-residue-representation (BRR) to generate super 2+1 dimensional integrable systems. Using these systems, we can deduce a super 2+1 dimensional AKNS hierarchy, which can be reduced to a super 2+1 dimensional nonlinear Schrödinger equation. In particular, two main results are obtained. One of them is a set of super 2+1 dimensional integrable couplings. The other one is a 2+1 dimensional diffusion equation. The Hamiltonian structure of the super 2+1 dimensional hierarchy is derived by using the super-trace identity.