A Lattice Hierarchy with a Free Function and Its Reductions to the Ablowitz-Ladik and Volterra Hierarchies

By embedding a free function into a compatible zero curvature equation, we propose a lattice hierarchy with the free function which still admits zero curvature representation. It is interesting that the hierarchy can reduce the Ablowitz-Ladik hierarchy, the Volterra hierarchy and a new hierarchy by properly choosing the embedded function. Moreover, the new hierarchy is integrable in Liouville’s sense and possess multi-Hamiltonian structure.     

Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

The noncommutative Toda hierarchy is studied with the help of Moyal deformation by a reduction on the non-commutative two dimensional Toda hierarchy. Further we generalize the noncommutative Toda hierarchy to the extended noncommutative Toda hierarchy. To survey on its integrability, we construct the bi-Hamiltonian structure and noncommutative conserved densities of the extended noncommutative Toda hierarchy by means of the R-matrix formalism. This extended noncommutative Toda hierarchy can be reduced to the extended multicomponent Toda hierarchy, extended Z_N?-Toda hierarchy, extended Toda hierarchy respectively by reductions on Lie algebras.     

New integrable lattice hierarchies

In this Letter we give a new integrable four-field lattice hierarchy, associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. We consider reductions of our hierarchy, which also of course admit discrete zero curvature representations, in detail. We find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We also obtain here a new integrable two-field lattice hierarchy, to which we give the name of Suris lattice hierarchy, since the first equation of this hierarchy has previously been given by Suris. The Hamiltonian structure of the Suris lattice hierarchy is obtained by means of a trace identity formula.     

Vector Loop Algebra and Its Applications to Tu Hierarchy

A vector loop algebra and its extended loop algebra are proposed, which are devoted to obtaining the Tu hierarchy. By making use of the extended trace identity, the Harniltonian structure of the Tu hierarchy is constructed. Furthermore, we apply the quadratic-form identity to the integrable coupling system of the Tu hierarchy.     

The super-classical-Boussinesq hierarchy and its super-Hamiltonian structure

Based on the constructed Lie superalgebra,the super-classical-Boussinesq hierarchy is obtained.Then,its superHamiltonian structure is obtained by making use of super-trace identity.Furthermore,the super-classical-Boussinesq hierarchy is also integrable in the sense of Liouville.     

A New Negative Discrete Hierarchy and Its N-Fold Darboux Transformation

Starting from a matrix discrete spectral problem, we derive a negative discrete hierarchy. It is shown that the hierarchy is integrable in the Liouville sense and possesses a bi-Hamiltonian structure. Furthermore, its N-fold Darboux transformation is established with the help of gauge transformation of Lax pair. As an application of the Darboux transformation, some new exact solutions for a discrete equation in the negative hierarchy are obtained.     

A lattice hierarchy and its continuous limits

By introducing a discrete spectral problem, we derive a lattice hierarchy which is integrable in Liouville's sense and possesses a multi-Hamiltonian structure. It is show that the discrete spectral problem converges to the well-known AKNS spectral problem under a certain continuous limit. In particular, we construct a sequence of equations in the lattice hierarchy which approximates the AKNS hierarchy as a continuous limit.     

A Hamiltonian structure associated with a matrix spectral problem of arbitrary-order

Starting from a specific matrix iso-spectral problem, an associated hierarchy of multi-component Hamiltonian equations is constructed, based on zero curvature equations. The key point is to choose appropriate time parts of Lax pairs which can yield evolution equations, and the existence of a Hamiltonian structure for the obtained hierarchy is established by means of the trace identity. An example with five components is computed, along with its Hamiltonian structure.     

New block matrix spectral problem and Hamiltonian structure of the discrete integrable coupling system

In [W.X. Ma, J. Phys. A: Math. Theor. 40 (2007) 15055], Prof. Ma gave a beautiful result (a discrete variational identity). In this Letter, based on a discrete block matrix spectral problem, a new hierarchy of Lax integrable lattice equations with four potentials is derived. By using of the discrete variational identity, we obtain Hamiltonian structure of the discrete soliton equation hierarchy. Finally, an integrable coupling system of the soliton equation hierarchy and its Hamiltonian structure are obtained through the discrete variational identity.     

An so (3, ℝ) Counterpart of the Dirac Soliton Hierarchy and its Bi-Integrable Couplings

We derive a counterpart hierarchy of the Dirac soliton hierarchy from zero curvature equations associated with a matrix spectral problem from so (3, ?). Inspired by a special class of non-semisimple loop algebras, we construct a hierarchy of bi-integrable couplings for the counterpart soliton hierarchy. By applying the variational identities which cope with the enlarged Lax pairs, we generate the corresponding Hamiltonian structure for the hierarchy of the resulting bi-integrable couplings. To show Liouville integrability, infinitely many commuting symmetries and conserved densities are presented for the counterpart soliton hierarchy and its hierarchy of bi-integrable couplings.     

带有层级结构的复杂网络级联失效模型

针对现实世界的网络中普遍存在的层级结构建立一个级联失效模型, 该模型可用于优化金融、物流网络设计. 选择的层级网络模型具有树形骨架和异质的隐含连接, 并且骨架中每层节点拥有的分枝数服从正态分布. 级联失效模型中对底层节点的打击在不完全信息条件下进行, 也即假设打击者无法观察到隐含连接. 失效节点的负载重分配考虑了层级异质性, 它可以选择倾向于向同级或高层级完好节点分配额外负载. 仿真实验表明, 层级网络的拓扑结构随连接参数变化逐渐从小世界网络过渡到随机网络. 网络级联失效规模随隐含连接比例呈现出先增加后降低的规律. 负载重分配越倾向于高层级节点, 网络的抗毁损性越高. 同时, 由于连接参数会改变隐含连接在不同层级之间的分布, 进而对网络的抗毁损性产生显著影响, 为了提高网络抗毁损能力, 设计网络、制定管理控制策略时应合理设定连接参数. 关键词： 复杂网络 级联失效 层级结构     

A Hierarchy of Integrable Lattice Soliton Equations and New Integrable Symplectic Map

Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Bäcklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.     

A Direct Method of Hamiltonian Structure

A direct method of constructing the Hamiltonian structure of the soliton hierarchy with self-consistent sources is proposed through computing the functional derivative under some constraints. The Hamiltonian functional is related with the conservation densities of the corresponding hierarchy. Three examples and their two reductions are given.     

Modular Classes of Poisson–Nijenhuis Lie Algebroids

The modular vector field of a Poisson–Nijenhuis Lie algebroid A is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian A-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson–Nijenhuis structure.      

Geometric effects on complex network structure in the cortex

It is shown that homogeneous, short-range, two-dimensional (2D) cortical connectivity, without modularity, hierarchy, or other specialized structure, reproduces key observed properties of cortical networks, including low path length, high clustering and modularity index, and apparent hierarchical block-diagonal structure in connection matrices. Geometry strongly influences connection matrices, implying that simple interpretations of connectivity measures as reflecting specialized structure can be misleading: Such apparent structure is seen in strictly uniform, locally connected architectures in 2D. Geometry is thus a proxy for function, modularity, and hierarchy and must be accounted for when structural inferences are made.     

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

We construct nonlinear super integrable couplings of the super integrable Dirac hierarchy based on an enlarged matrix Lie superalgebra. Then its super Hamiltonian structure is furnished by super trace identity. As its reduction, we gain the nonlinear integrable couplings of the classical integrable Dirac hierarchy.     

A Hierarchy of Nonlinear Lattice Soliton Equations and Its Darboux Transformation

A hierarchy of nonlinear lattice soliton equations is derived from a new discrete spectral problem. The Hamiltonian structure of the resulting hierarchy is constructed by using a trace identity formula. Moreover, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations. The exact solutions are given by applying the Darboux transformation.     

The multicomponent （2＋1）-dimensional Glachette-Johnson （GJ） equation hierarchy and its super-integrable coupling system

This paper presents a set of multicomponent matrix Lie algebra, which is used to construct a new loop algebra A^-M. By using the Tu scheme, a Liouville integrable multicomponent equation hierarchy is generated, which possesses the Hamiltonian structure. As its reduction cases, the multicomponent （2＋1）-dimensional Glachette-Johnson （G J） hierarchy is given. Finally, the super-integrable coupling system of multicomponent （2＋1）-dimensional GJ hierarchy is established through enlarging the spectral problem.     

A new generalized fractional Dirac soliton hierarchy and its fractional Hamiltonian structure

Based on the differential forms and exterior derivatives of fractional orders,Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation.We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure.The method can be generalized to the other fractional soliton hierarchy.