Lax representations and integrable time discretizations of the DDKdV, DDmKdV, and DDHOKdV

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 KdV 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.