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Two coupled integrable hierarchies associated with a proper isospectral problem are obtained. These hierarchies possess hereditary symmetry structure but each hierarchy is not Hamiltonian.

From a proper 2×2 discrete isospectral problem, the relativistic Volterra lattice introduced by Suris and Ragnisco is rederived. So, a new Lax matrix for the relativistic Volterra lattice is given....

From a proper 2 × 2 discrete isospectral problem, a new differential-difference integrable equation in Lax sense is proposed by a discrete zero curvature equation. The DDKdV (differential-difference DdV equation) proposed by Ohta and Hirota and DDCDGKS (DD Caudrey-Dodd-Gibbon-Kotera-Sawada equation) are rederived. Some other new discrete KdV equations, discrete mKdV equations and discrete high order KdV equations which converge to the corresponding continuous soliton equations in the continuum limit are obtained. Integrable time discretizations of the DDKdV, DDmKdV (differential-difference mKdV equation) and DDHOKdV (differential-difference high order KdV equations) are given.

From a proper 2 × 2 discrete isospectral problem, a new integrable lattice soliton system is proposed. Integrable discretizations of a general Toda-type lattice soliton equation associated with the discrete isospectral problem are established. The Lagrangian and Newtonian forms of integrable discretizations of Toda-type lattice equations which occur in the literature are given uniformly and some new integrable discretizations of the Toda-type lattice are obtained.

A (2 m + 1)th-order generalized KdV equation is considered, where m is a positive integer. Four new explicit travelling solitary-wave solutions are obtained for the case m = 4. The case of arbitrary m ≥ 3 is considered; it is shown that, subject to certain restrictions on the coefficients in the equation, there are always at least two sech-polynomial type solutions.

Some new nonlinear differential-difference integrable hierarchies associated with a properly discrete spectral problem are obtained. The special cases of the proposed differential-difference integrable hierarchies are discussed. The Ablowitz-Ladik discretization of the NLS equation, the discrete modified KdV equation, the modified Toda lattice, the relativistic Toda lattice, and some other lattice soliton equations obtained by Suris are the special examples of novel differential-difference integrable hierarcies.

A new Lax integrable hierarchy associated with a properly spectral problem is obtained. Using the trance identity method, the Hamiltonian form of the proposed Lax hierarchy is also given.

Some new Lax integrable hierarchies associated with 3 × 3 matrix spectral problems and their bi-Hamiltonian structures are obtained.

We study the extrapolation method for the numerical solution of elliptic boundary value problems on corner domains. Most papers on the extrapolation method for solving differential equations require smoothness assumptions on exact solutions, which do not hold for boundary value problems on corner domains. We conjecture that asymptotic error expansions exist as long as there is enough smoothness on the input data except the domain. This is confirmed by our theoretical result on asymptotic error expansions for model problems on a rectangle. Numerical examples suggest that similar results hold for problems on more general corner domains, although it seems very difficult to prove them.