Nonlinear Integrable Couplings Research Articles

Integrable couplings stemming from three-dimensional unital algebras

This study aims to explore the connection between integrable couplings and three-dimensional unital algebras. Expansions in these algebras generate integrable couplings, including nonlinear integrable couplings, and their corresponding Lax pairs and hereditary recursion operators possess specific structures, associated with non-semisimple Lie algebras. Applications to the KdV equation are explicitly presented.

Integrable Couplings and Two-Dimensional Unital Algebras

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schrödinger equations are extensively explored.

Several Isospectral and Non-Isospectral Integrable Hierarchies of Evolution Equations

By introducing a 3×3 matrix Lie algebra and employing the generalized Tu scheme, a AKNS isospectral–nonisospectral integrable hierarchy is generated by using a third-order matrix Lie algebra. Through a matrix transformation, we turn the 3×3 matrix Lie algebra into a 2×2 matrix case for which we conveniently enlarge it into two various expanding Lie algebras in order to obtain two different expanding integrable models of the isospectral–nonisospectral AKNS hierarchy by employing the integrable coupling theory. Specially, we propose a method for generating nonlinear integrable couplings for the first time, and produce a generalized KdV-Schrödinger integrable system and a nonlocal nonlinear Schrödinger equation, which indicates that we unite the KdV equation and the nonlinear Schrödinger equation as an integrable model by our method. This method presented in the paper could apply to investigate other integrable systems.

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.

The quasi-AKNS and quasi-[formula omitted] type AKNS hierarchies as well as a gallery of simple Lie algebras contained in [formula omitted

This paper provides a new representation of sl(3;ℂ), i.e., a high-dimensional special linear Lie algebra A, which contributes to the generation of a series of simple Lie algebras owning different bases with inequable commutators. Their counterparts are obtained in succession by using some special invertible linear transformations. As applications, the quasi-AKNS and quasi-so(3) type AKNS Hamiltonian hierarchies as well as their nonlinear continuous integrable couplings with the Hamiltonian structures are established, respectively.

Some novel soliton solution, breather solution and Darboux transformation for a generalized coupled Toda soliton hierarchy

A few of discrete integrable coupling systems(DICSs) of previous papers are linear discrete integrable couplings(LDICS). We take a special matrix Lie algebra system(non-semisimple) to construct the Lax pairs, and establish a method for deriving the nonlinear discrete integrable coupling systems(NDICS). From the Lax pairs of the generalized Toda(G-Toda) spectral problem, we can derive a novel NDICS, which is a real NDICS. For the obtained lattice integrable coupling equation, we establish a Darboux transformation (DT) with 4 × 4 Lax pairs, and apply the gauge transformation to a specific equation, then the explicit solutions of the lattice integrable coupling equation are given, which contains discrete soliton solution, breather solution and rogue wave solution. Furthermore, we can derive the discrete explicit solutions with free parameters to depict their dynamic behaviors.

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.

Two New Integrable Hierarchies and Their Nonlinear Integrable Couplings

By introducing an invertible linear transform, a new Lie algebra G is obtained from the Lie algebra H. Making use of the compatibility conditions of the respective isospectral problems, a generalized NLS-MKdV hierarchy and a new integrable soliton hierarchy are achieved by using the trace identity or the variational identity. Then, two special non-semisimple Lie algebras and are explicitly conducted. As an application, the nonlinear continuous integrable couplings of the obtained integrable systems as well as their bi-Hamiltonian structures are established, respectively.

Nonlinear integrable couplings of a generalized super Ablowitz‐Kaup‐Newell‐Segur hierarchy and its super bi‐Hamiltonian structures

In this paper, a new generalized 5×5 matrix spectral problem of Ablowitz‐Kaup‐Newell‐Segur type associated with the enlarged matrix Lie superalgebra is proposed, and its corresponding super soliton hierarchy is established. The super variational identities are used to furnish super Hamiltonian structures for the resulting super soliton hierarchy.

Dynamic behavior of traveling wave solutions for new couplings of the Burgers equations with time-dependent variable coefficients

In this paper, we develop the nonlinear integrable couplings of Burgers equations with time-dependent variable coefficients. A new simplified bilinear method is used to obtain new multiple-kink solutions and multiple-singular-kink solutions for this system. The proposed system is a generalization model in ocean dynamics, plasma physics and nonlinear lattice. The effects of time-variable coefficients on the velocity, phase and amplitude are given. The solitonic propagation and collision are discussed by the graphical analysis and characteristic-line method.

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.

Nonlinear Integrable Couplings of the Kaup-Newell Hierarchy

In this paper, a six-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which an isospectral problem is established. By zero curvature equations, we obtain the nonlinear integrable couplings of the Kaup-Newell (KN) hierarchy.

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.

Nonlinear Integrable Couplings of the Kaup-Newell Hierarchy

In this paper, a six-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which an isospectral problem is established. By zero curvature equations, we obtain the nonlinear integrable couplings of the Kaup-Newell (KN) hierarchy.

Nonlinear Super Integrable Couplings of Super Classical Boussinesq Hierarchy

Enlargement of Lie super algebra B(0, 1) was given firstly. Then nonlinear super integrable couplings of the super classical Boussinesq hierarchy based upon this enlarged matrix Lie super algebra were constructed secondly. And its super Hamiltonian structures were established by using super trace identity thirdly. As its reduction, special integrable couplings of classical Boussinesq hierarchy were obtained finally.

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.

Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced Lie algebras, denoted by R 6, R 8 and R 1 6 ,R 2 6 , respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R 6 whose compatibility gives rise to an integrable hierarchy with 4-potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R 1 6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R 2 6 , we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.

A nonlinear integrable couplings of C-KdV soliton hierarchy and its infinite conservation laws

A kind of nonlinear integrable couplings of C-KdV soliton hierarchy is constructed, then we present the infinitely many conservation laws for the nonlinear integrable couplings of the C-KdV soliton hierarchy.

Two Expanding Integrable Models of the Geng-Cao Hierarchy

As far as linear integrable couplings are concerned, one has obtained some rich and interesting results. In the paper, we will deduce two kinds of expanding integrable models of the Geng-Cao (GC) hierarchy by constructing different 6-dimensional Lie algebras. One expanding integrable model (actually, it is a nonlinear integrable coupling) reduces to a generalized Burgers equation and further reduces to the heat equation whose expanding nonlinear integrable model is generated. Another one is an expanding integrable model which is different from the first one. Finally, the Hamiltonian structures of the two expanding integrable models are obtained by employing the variational identity and the trace identity, respectively.

Nonlinear Super Integrable Couplings of Super Classical-Boussinesq Hierarchy

Nonlinear integrable couplings of super classical-Boussinesq 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 the classical integrable hierarchy were obtained.