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We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotkaâ€"Volterra, Toda lattice (TL), and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional TL equations. Each of the delay-difference and delay-differential equations has the N -soliton solution, which depends on the delay parameter and converges to an N -soliton solution of a known soliton equation as the delay parameter approaches 0. [ABSTRACT FROM AUTHOR]