Super Integrable Hierarchy Research Articles

A multi-component super integrable Dirac hierarchy

We propose a method for generating higher-dimensional nonisospectral super integrable coupling hierarchies associated with a new type of higher-dimensional Lie superalgebra. As applications, we introduce a coupled and a multi-component super Dirac spectral problems respectively, and thus deduce a coupled generalized super integrable Dirac hierarchy and a multi-component generalized super integrable Dirac hierarchy. Additionally, we discuss the super Hamiltonian structures of the resulting higher-dimensional super Dirac hierarchies.

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.

2＋1 dimensional nonisospectral super integrable hierarchies associated with a class of extended Lie superalgebras

Based on a class of extended Lie superalgebras that we constructed, we present a method for generating 2＋1 dimensional nonisospectral super integrable hierarchies. By considering some applications, we derive three kinds of 2＋1 dimensional nonisospectral super Dirac integrable hierarchies and AKNS integrable hierarchies. Moreover, super Hamiltonian structures of these 2＋1 dimensional super integrable hierarchies are obtained.

Two Isospectral-Nonisospectral Super-Integrable Hierarchies and Related Invariant Solutions

In this article, we adopt two kinds of loop algebras corresponding to the Lie algebra B(0,1) to introduce two line spectral problems with different numbers of even and odd superfunctions. Through generalizing the time evolution λt to a polynomial of λ, two isospectral-nonisospectral super integrable hierarchies are derived in terms of Tu scheme and zero-curvature equation. Among them, the first super integrable hierarchy is further reduced to generalized Fokker–Plank equation and special bond pricing equation, as well as an explicit super integrable system under the choice of specific parameters. More specifically, a super integrable coupled equation is derived and the corresponding integrable properties are discussed, including the Lie point symmetries and one-parameter Lie symmetry groups as well as group-invariant solutions associated with characteristic equation.

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 super D-Kaup-Newell hierarchy and its nonlinear integrable coupling with self-consistent sources

How to construct new super integrable equation hierarchy is an important problem. In this paper, a new Lax pair is proposed and the super D-Kaup-Newell hierarchy is generated, then a nonlinear integrable coupling of the super D-Kaup-Newell hierarchy is constructed. The super Hamiltonian structures of coupling equation hierarchy is derived with the aid of the super variational identity. Finally, the self-consistent sources of super integrable coupling hierarchy is established. It is indicated that this method is a straightforward and efficient way to construct the super integrable equation hierarchy.

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.

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.

Darboux transformations for super-Schrödinger equation, super-Dirac equation and their exact solutions

The Darboux transformation (DT) for the super-integrable hierarchy has an essential difference from the general system. As we know, the super-integrable soliton equation hierarchies with four potentials are discussed. Starting from the spectral problems of super-AKNS hierarchy and super-Dirac hierarchy, a DT method for two super-integrable hierarchies is constructed, which is more complex than the general integrable system. Soliton solutions of super-Schrodinger equation and super-Dirac equation are presented by using DT, which contain some bright, dark and breather wave soliton solutions. Then, the properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients.

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.

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.

A Few Super-Integrable Hierarchies and Some Reductions, Super-Hamiltonian Structures

Super-integrable equations provide some important physical models. In the paper, we have introduced two bases of the super Lie algebra B(0, 1). One of them is used to link three super hierarchies of evolution equations under the 0-curvature equations. We have constructed its two kinds of super loop algebras. By employing the first loop algebra, an isospectral problem was introduced for which a super (1 + 1)-dimensional integrable hierarchy is obtained, whose super Hamiltonian structure was derived from the super-trace identity proposed by Tu et al. As the reduction cases, we have obtained two different explicit super Schrödinger and mKdV systems. Further reduced equations are the well-known nonlinear Schrödinger equation, the super KdV equations and the mKdV equation. By employing the second loop algebra, we have obtained a new super integrable hierarchy containing 5 even superfunctions and 4 odd ones, which possesses super bi-Hamiltonian structure. As the reduction case, we have obtained an explicit super integrable system which is further reduced to 3 super integrable coupled equations, especially we have got a super coupled constrained equation with sources with respect to the odd superfunctions. Finally, we have used the TAH scheme to obtain a super (2+1)-dimensional hierarchy of evolution equations, whose Hamiltonian structure was generated by a new trace identity developed by us in the paper, which are new results, to the best of our knowledge.

Bargmann Symmetry Constraint and Binary Nonlinearization of a Super Integrable Hierarchy

Bargmann Symmetry Constraint and Binary Nonlinearization of a Super Integrable Hierarchy

A Super Integrable Hierarchy and Its Nonlinear Super Integrable Hamiltonian Couplings

Basing on an existing Lie super algebra, we introduce a kind of a new Lie super algebra G and establish two isospectral Lax matrices. By taking use of the Tu scheme and the super trace identity, a super integrable hierarchy along with Hamiltonian structure is deduced, from which some evolution equations are obtained, including the well-known mKdV equation. Then, we employ the Lie super algebra G to extend a new higher-dimensional Lie super algebra G ¯ , from which the nonlinear super integrable couplings of the super integrable hierarchy are given. Moreover, the super trace identity over the corresponding loop super algebra is used to furnish the super Hamiltonian structure for the resulting nonlinear super integrable couplings.

A New Super Extension of Dirac Hierarchy

We derive a new super extension of the Dirac hierarchy associated with a3×3matrix super spectral problem with the help of the zero-curvature equation. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy.

Super-Hamiltonian Structures and Conservation Laws of a New Six-Component Super-Ablowitz-Kaup-Newell-Segur Hierarchy

A six-component super-Ablowitz-Kaup-Newell-Segur (-AKNS) hierarchy is proposed by the zero curvature equation associated with Lie superalgebras. Supertrace identity is used to furnish the super-Hamiltonian structures for the resulting nonlinear superintegrable hierarchy. Furthermore, we derive the infinite conservation laws of the first two nonlinear super-AKNS equations in the hierarchy by utilizing spectral parameter expansions. PACS: 02.30.Ik; 02.30.Jr; 02.20.Sv.

Conservation laws and self-consistent sources for a super KN hierarchy

In the present paper, a super-extension of the Kaup–Newell (KN) hierarchy is proposed by matrix Lie super algebras and the super KN hierarchy with self-consistent sources is established. Furthermore, infinitely many conservation laws of the super integrable hierarchy are presented. The methods derived by us can be generalized to other nonlinear equation hierarchies.

The conservation laws and self-consistent sources for a super-Yang hierarchy

In the present paper, a super-extension of the Yang hierarchy is proposed by super-matrix Lie algebras, and the super-Yang hierarchy with self-consistent sources is established. Furthermore, we establish infinitely many conservation laws of the super-integrable hierarchy. The methods presented by us can be generalized to other nonlinear equation hierarchies with self-consistent sources.