Noncommutative AKNS Equation Hierarchy and Its Integrable Couplings with Kronecker Product

We present a noncommutative version of the Ablowitz-Kaup-Newell-Segur （AKNS） equation hierarchy, which possesses the zero curvature representation. Furthermore, we derive the noncommutative AKNS equation from the noncommutative （anti-）self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, the integrable coupling system of the noncommutative AKNS equation hierarchy is constructed by using the Kronecker product.     

A New Integrable Lattice Hierarchy and Its Two Discrete Integrable Couplings

A new and efficient way is presented for discrete integrable couplings with the help of two semi-direct sum Lie algebras. As its applications, two discrete integrable couplings associated with the lattice equation are worked out. The approach can be used to study other discrete integrable couplings of the discrete hierarchies of solition equations.     

Lie Algebra and Lie Super Algebra for Integrable Couplings of C-KdV Hierarchy

Based on the constructed Lie algebra and Lie super algebra, the integrable couplings and super-integrable couplings of the C-KdV hierarchy are obtained respectively. Furthermore, its super-Hamiltonian structures are presented by using super-trace identity.     

Decomposition for a 2+1-Dimensional Discrete Integrable Model

A 2＋1-dimensional discrete is presented, which is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems, with aid of the nonlineaxization of Lax pairs. The system is completely integrable in the Liouville sense.     

Approximate Symmetry Reduction to the Perturbed One-Dimensional Nonlinear Schrödinger Equation

The one-dimensional nonlinear Schrödinger equation with a perturbation of polynomial type is considered. Using the approximate symmetry perturbation theory, the approximate symmetries and approximate symmetry reduction equations are obtained.     

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.     

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.     

Integrable Rosochatius Deformations of the Restricted cKdV Flows

Three novel finite-dimensional integrable Hamiltonian systems of Rosochatius type and their Lax representations are presented. We make a deformation for the Lax matrbces of the Neumann type, the Bargmann type and the high-order symmetry type of restricted cKdV flows by adding an additional term and then prove that this kind of deformation does not change the r-matrix relations. Finally the new integrable systems are generated from these deformed Lax matrices.     

New Periodic Solution to Jacobi Elliptic Functions of a (2+1)-Dimensional BKP Equation and a Generalized Klein-Gordon Equation

With the help of the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to obtain the Jacobi doubly periodic wave solutions of the (2+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation and the generalized Klein-Gordon equation. The method is also valid for other (1+1)-dimensional and higher dimensional systems.     

New Integrable and Linearizable Nonlinear Difference Equations

A systematic investigation to derive nonlinear lattice equations governed by partial difference equations (PΔΔE) admitting specific Lax representation is presented. Further it is shown that for a specific value of the parameter the derived nonlinear PΔΔE's can be transformed into a linear PΔΔE's under a global transformation. Also it is demonstrated how to derive higher order ordinary difference equations (OΔE) or mappings in general and linearizable ones in particular from the obtained nonlinear PΔΔE's through periodic reduction. The question of measure preserving property of the obtained OΔE's and the construction of more than one integrals (or invariants) of them is examined wherever possible.     

From Yang-Baxter Maps to Integrable Recurrences

Starting from known solutions of the functional Yang-Baxter equations, we construct a series of nonautonomous integrable recurrences, “median graphs”, and give their explicit solution.     

Integrable coupling system of fractional soliton equation hierarchy

In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained.     

Higher-Dimensional Integrable Systems Induced by Motions of Curves in Affine Geometries

We discuss the motions of curves by introducing an extra spatial variable or equivalently, moving surfaces in arffine geometries. It is shown that the 2 ＋1-dimensional breaking soliton equation and a 2 ＋ 1-dimensional nonlinear evolution equation regarded as a generalization to the 1 ＋ 1-dimensional KdV equation arise from such motions.     

Multi-Soliton Solutions and Integrable Discretization for a Coupled Modified Volterra Lattice Equation

In this paper, we study a coupled modified Volterra lattice equation which is an integrable semidiscrete version of the coupled KdV and the coupled mKdV equation. By using the Darboux transformation, we obtain its new explicit solutions including multi-soliton and multi-positon. Furthermore, an integrable discretization of the coupled modified Volterra lattice equation is constructed.     

Multi-Soliton Solutions and Integrable Discretization for a Coupled Modified Volterra Lattice Equation

In this paper, we study a coupled modified Volterra lattice equation which is an integrable semidiscrete version of the coupled KdV and the coupled mKdV equation. By using the Darboux transformation, we obtain its new explicit solutions including multi-soliton and multi-positon. Furthermore, an integrable discretization of the coupled modified Volterra lattice equation is constructed.     

A Hierarchy of Differential-Difference Equations and Their Integrable Couplings

Starting from a discrete spectral problem, the corresponding hierarchy of nonlinear differential-difference equation is proposed. It is shown that the hierarchy possesses the bi-Hamiltionian structures. Further, two integrable coupling systems for the hierarchy are constructed through enlarged Lax pair method.     

Integrable Couplings of the Boiti-Pempinelli-Tu Hierarchy and Their Hamiltonian Structures

Integrable couplings of the Boiti-Pempinelli-Tu hierarchy are constructed by a class of non-semisimple block matrix loop algebras. Further, through using the variational identity theory, the Hamiltonian structures of those integrable couplings are obtained. The method can be applied to obtain other integrable hierarchies.     

Integrable Couplings of the Coupled Burgers Hierarchy

In this letter, a new loop algebra G is constructed, from which a new isospectral problem is established. It follows that integrable couplings of the well-known coupled Burgers hierarchy are obtained.     

Integrable Couplings of the Coupled Burgers Hierarchy

In this letter, a new loop algebra G is constructed, from which a new isospectral problem is established. It follows that integrable couplings of the well-known coupled Burgers hierarchy are obtained.     

Integrable Rosochatius Deformations for an Integrable Couplings of CKdV Equation Hierarchy

We propose a method to construct the integrable Rosochatius deformations for an integrable couplings equations hierarchy. As applications, the integrable Rosochatius deformations of the coupled CKdV hierarchy with self-consistent sources and its Lax representation are presented.