The two-dimensional autonomous cellular neural networks (CNNs) having one layer or two layers of memristor coupling can exhibit many interesting nonlinear waves and bifurcation phenomena. In this paper, we study the nonlinear waves (solitons) in the one-dimensional CNN difference equations. From our computer simulations, we found that the CNN difference equations can exhibit many interesting behaviors. The most remarkable thing is that the first-order linear CNN difference equation can exhibit a train of solitary waves, if the initial condition is given by the unit step function. Furthermore, the second-order linear CNN difference equation can exhibit soliton-like behavior, if the initial condition is given by a pulse wave. That is, the solitary waves pass through one another and emerge from the collision. Furthermore, the solution exhibits the area-preserving behavior, and it returns exactly to its initial state (the recurrence of the initial state). In the case of the nonlinear CNN difference equation, we observed the following interesting behaviors. In the Korteweg-de Vries CNN difference equation, the three-dimensional plot of the interaction of the solitary waves looks like a chicken cockscomb. In the Toda lattice CNN difference equation, a train of solitary waves with a negative amplitude interact with a train of solitary waves with a positive amplitude, and they emerge from the collisions. Furthermore, after a certain period of time, the solution breaks down. In the Sine-Gordon CNN difference equation, the solution moves at constant speed, and it emerges from the collision. Furthermore, the solution returns the state which is roughly similar to the initial state. In the memristor CNN difference equations, the three-dimensional plots of solitary waves exhibit more complicated (chaotic or distorted) behavior.