A hybrid type of soliton equations with self-consistent sources: KP and Toda cases

Source generation procedure is applied to construct a hybrid type of soliton equations with self-consistent sources (SESCSs). The examples include the KP equation with self-consistent sources (KPESCS) and two-dimensional TodaESCS. One typical feature for this hybrid type of SESCSs is that soliton solutions of these new systems contain arbitrary functions of a linear combination of two independent variables, which is different from the normal SESCSs where soliton solutions only contain arbitrary functions of one independent variable. Whatʹs more, the obtained two hybrid SESCSs can be reduced to two different simpler SESCSs respectively.