Gauge transformations between three-component KP and three-component mKP hierarchies

We examine the gauge transformations between the three-component Kadomtsev–Petviashvili (KP) hierarchy and the three-component modified Kadomtsev–Petviashvili (mKP) hierarchy. We introduce the auto-Bäcklund transformation of the three-component KP hierarchy and describe the Miura transformation of the three-component KP and mKP hierarchies, that is, \(k=0\to k=1\) . Additionally, the auto-Bäcklund transformation of the three-component mKP hierarchy is also given. This provides more powerful evidence that the gauge transformations generate Miura and auto-Bäcklund transformation on the eigenfunctions of the three-component KP and mKP hierarchies.