An algebraic-geometric construction of "lump" solutions of the KP1 equation

Abstract

In this note, we show how certain everywhere-regular real rational function solutions of the KP1 equation ("multi-lumps") can be constructed via the polynomial analogs of theta functions from singular rational curves with cusps. We use two methods, one direct and the other producing a degeneration of the well-understood soliton solutions from nodal singular curves. The second approach can be seen as a variation on the long-wave limit technique of Ablowitz and Satsuma, as developed by Zhang, Yang, Li, Guo, and Stepanyants. We present an explicit example of a three-lump solution constructed via the polynomial analog of the theta function from a rational curve with two cuspidal singular points, each with semigroup $\langle 2,5\rangle$. (In the theory of curve singularities, these are known as $A_4$ double points.) We conjecture that these ideas will generalize to give similar $M$-lump solutions with $M = \frac{N(N+1)}{2}$ for $N > 2$ starting from rational curves with two singular points with semigroup $\langle 2,2N+1\rangle$ ($A_{2N}$ double points). We also show a five-lump solution obtained from a curve with two cusps with semigroup $\langle 3,4\rangle$. Similar solutions have been constructed by other methods previously; our contribution is to show how they arise from the algebraic-geometric setting by considering singular curves with several cusps, as in previous work of Agostini, Celik, and Little.