Cauchy matrix structure of the Mel'nikov model of long-short wave interaction

We propose a systematic method to construct the Mel'nikov model of long-short wave interactions, which is a special case of the Kadomtsev-Petviashvili (KP) equation with self-consistent sources (KPSCS). We show details how the Cauchy matrix approach applies to Mel'nikov's model which is derived as a complex reduction of the KPSCS. As a new result we find that in the dispersion relation of a 1-soliton there is an arbitrary time-dependent function that has previously not reported in the literature about the Mel'nikov model. This function brings time variant velocity for the long wave and also governs the short-wave packet. The variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated.