Abstract: With respect to integrable partial differential equations, gauge transformations of Lax pairs are critical tools that can be used to extend integrable equations. We investigate gauge transformations for discrete integrable equations. By applying twice gauge transformations to the Lax pair of the discrete Korteweg—de Vries (KdV) equation, we obtain discrete modified KdV and discrete modified KdV-Ⅱ equations. Then, by introducing the potential variables, we obtain potential forms of those two equations, which are proven to satisfy the three-dimensional consistency property.

Key words: gauge transformation, 3D consistency, discrete KdV equation