Soliton Resonances for the MKP-II

Using the second flow (derivative reaction-diffusion system) and the third one of the dissipative $SL(2,\mathbb R)$ Kaup–Newell hierarchy, we show that the product of two functions satisfying those systems is a solution of the modified Kadomtsev–Petviashvili equation in $2+1$ dimensions with negative dispersion (MKP-II). We construct Hirota's bilinear representations for both flows and combine them as the bilinear system for the MKP-II. Using this bilinear form, we find one- and two-soliton solutions for the MKP-II. For special values of the parameters, our solution shows resonance behavior with the creation of four virtual solitons. Our approach allows interpreting the resonance soliton as a composite object of two dissipative solitons in $1+1$ dimensions.