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.

我们提出了一种系统的方法来构造长短波相互作用的梅尔尼科夫模型，这是带有自洽源的Kapdomtsev-Petviashvili（KP）方程的特例。我们展示了柯西矩阵方法如何应用于梅尔尼科夫模型的细节，该模型是作为KPSCS的复杂约简而得出的。作为一个新的结果，我们发现在1-孤子的色散关系中，存在一个随时间变化的函数，该函数以前在Mel'nikov模型的文献中没有报道过。此功能为长波带来时变速度，并控制短波数据包。示出了由色散关系中的时间自由性引起的波的相互作用的多样性。