A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

A nonisospectral integrable model of AKNS hierarchy and KN hierarchy, as well as its extended system

In this paper, we first introduce a nonisospectral problem associate with a loop algebra. Based on the nonisospectral problem, we deduce a nonisospectral integrable hierarchy by solving a nonisospectral zero curvature equation. It follows that the standard AKNS hierarchy and KN hierarchy are obtained by reducing the resulting nonisospectral hierarchy. Then, the Hamiltonian system of the resulting nonisospectral hierarchy is investigated based on the trace identity. Additionally, an extended integrable system of the resulting nonisospectral hierarchy is worked out based on an expanded higher-dimensional Loop algebra.

An integrable soliton hierarchy associated with the Boiti–Pempinelli–Tu spectral problem

Based on the Tu scheme [G.-Z. Tu, J. Math. Phys. 30 (1989) 330], we construct a counterpart of the Boiti–Pempinelli–Tu soliton hierarchy from a matrix spectral problem associated with the Lie algebra [Formula: see text], and formulate Hamiltonian structures for the resulting soliton equations by means of the trace identity. We then show that the newly presented equations possess infinitely many commuting symmetries and conservation laws. Finally, we derive the well-known combined KdV-mKdV equation from the new hierarchy.

Algebro-geometric constructions of the Heisenberg hierarchy

The Heisenberg hierarchy and its Hamiltonian structure are derived respectively by virtue of the zero-curvature equation and the trace identity. With the help of the Lax matrix, we introduce an algebraic curve ${\mathcal{K}}_{n}$ of arithmetic genus n, from which we define meromorphic function ϕ and straighten out all of the flows associated with the Heisenberg hierarchy under the Abel–Jacobi coordinates. Finally, we achieve the explicit theta function representations of solutions for the whole Heisenberg hierarchy as a result of the asymptotic properties of ϕ.

The Cry of the World

Introduces some key concepts: hybridity, ‘Relation’, the relation between oral and written language, creolization, the chaos-world, multilingualism and ‘opacity’ (i.e., we do not need to understand the other in order to relate to him/her.) From now on, we can all hear the cry of the world, i.e. we are conscious of struggles in faraway places, and we live in ‘common places’ that we are learning to share. Glissant contrasts the ‘system’ with its positive alternative, the ‘trace’. Identity is recast as a relational ‘rhizome’ (cf Deleuze and Guattari), rather than a single self-sufficient ‘root’. (p.11). He stresses the importance of defending languages that are in danger of disappearing, but also discusses the virtues of translation. He describes the founding of the International Writers Parliament in Strasbourg.

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.

An extension of the coupled derivative nonlinear Schrödinger hierarchy

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

Multi-Component Generalizations of Two Hierarchies Related to the Yajima-Oikawa Hierarchy

We present a multi-component modified Yajima-Oikawa hierarchy and a multi-component Novikov hierarchy. We get generalized Hamiltonian structures for the two hierarchies with the aid of trace identity.

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

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 $n$-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.

Three-Component Generalisation of Burgers Equation and Its Bi-Hamiltonian Structures

A hierarchy of three-component generalisation of Burgers equation, which is associated with a 3×3 matrix eigenvalue problem, is generated by using the zero-curvature equation. By means of the trace identity, the bi-Hamiltonian structures of this hierarchy are constructed. Moreover, the infinite conservation laws for the hierarchy are obtained with the aid of spectral parameter expansion.