Regular line-solitons of modified KP system and vertex operators

In this paper, we study the effects of vertex operators on soliton solutions of the modified Kadomtsev–Petviashvili (mKP) equation based on the Grassmannian $Gr(N, M)$. We present a classification theorem for the solutions of the mKP equation and demonstrate the connection among vertex operators, theta functions, and the tau-function of the mKP equation. We also express the soliton solution on $Gr(N,M)$ of the mKP equation in terms of the $M$-theta function and free fermions. We show that the new tau-function obtained by the action of the vertex operator on the tau-function of the mKP equation can still form a solution of the mKP equation. We study all kinds of solutions of the mKP equation obtained through the action of vertex operators. We find that when the mKP equation solutions obtained via the action of the vertex operator satisfy the regularity condition, the chord diagram exhibits non-crossing between the graph related to the parameters of the vertex operators and the graph related to the parameters of the original solution.