Isospectral and Nonisospectral Lattice Hierarchies Associated with a Discrete Spectral Problem and Their Infinitely Many Conservation Laws

In this paper, based on the discrete zero curvature representation, isospectrai and nonisospectrai lattice hierarchies are proposed. By means of solving corresponding discrete spectral equations, we demonstrate the existence of infinitely many conservation laws for this two hierarchies and obtain the formulae of the corresponding conserved densities and associated fluxes.

Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

Starting from a new discrete spectral problem, the corresponding hierarchy of nonlinear lattice equations is proposed. It is shown that the lattice soliton hierarchy possesses the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, infinite conservation laws of the hierarchy are presented.

Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations

A fully discrete Jacobi-spherical harmonic spectral method is provided for the Navier-Stokes equations in a ball. Its stability and convergence are proved. Numerical results show efficiency of this approach. The proposed method is also applicable to other problems in spherical geometry.

WEAK ERROR ESTIMATES FOR TRAJECTORIES OF SPDEs UNDER SPECTRAL GALERKIN DISCRETIZATION

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation （analytic semigroup） and damped-driven wave equations （bounded semigroup） type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories： test functions are regular, but depend on the values of the process on the interval [0, T]. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function （It6 map） of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the It6 map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup： the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

AN ABLOWITZ-LADIK INTEGRABLE LATTICE HIERARCHY WITH MULTIPLE POTENTIALS

Within the zero curvature formulation,a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type.The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity.When the involved two potential vectors are scalar,all the resulting integrable lattice equations are reduced to the standard Ablowitz-Ladik hierarchy.

Hamiltonian System of New Nonlinear Lattice Equations

A 3-dimensional Lie algebra sμ（3） is obtained with the help of the known Lie algebra. Based on the sμ（3）, a new discrete 3 × 3 matrix spectral problem with three potentials is constructed. In virtue of discrete zero curvature equations, a new matrix Lax representation for the hierarchy of the discrete lattice soliton equations is acquired. It is shown that the hierarchy possesses a Hamiltonian operator and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.

A GLOBAL ALGORITHM IN THE NUMERICAL RESOLUTION OF THE VISCOUS/INVISCID COUPLED PROBLEM

This paper deals with the spectral approximation of an incompressible viscous/inviscid coupled model. An efficient Uzawa algorithm based on a new variational formulation is proposed. The generalization to the coupling between the Navier Stokes equations and the Euler equations is discussed.

Integrable Hierarchy Covering the Lattice Burgers Equation in Fluid Mechanics:N-fold Darboux Transformation and Conservation Laws

Burgers-type equations can describe some phenomena in fluids,plasmas,gas dynamics,traffic,etc.In this paper,an integrable hierarchy covering the lattice Burgers equation is derived from a discrete spectral problem.N-fold Darboux transformation(DT) and conservation laws for the lattice Burgers equation are constructed based on its Lax pair.N-soliton solutions in the form of Vandermonde-like determinant are derived via the resulting DT with symbolic computation,structures of which are shown graphically.Coexistence of the elastic-inelastic interaction among the three solitons is firstly reported for the lattice Burgers equation,even if the similar phenomenon for certern continuous systems is known.Results in this paper might be helpful for understanding some ecological problems describing the evolution of competing species and the propagation of nonlinear waves in fluids.

Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations

In this paper, based on a discrete spectral problem and the corresponding zero curvature representation,the isospectral and nonisospectral lattice hierarchies are proposed. An algebraic structure of discrete zero curvature equations is then established for such integrable systems. the commutation relations of Lax operators corresponding to the isospectral and non-isospectral lattice flows are worked out, the master symmetries of each lattice equation in the isospectral hierarchyand are generated, thus a τ-symmetry algebra for the lattice integrable systems is engendered from this theory.