2024/04/23 by John B. Little, Little, John B.
Mathematics · Physics and Astronomy · #14H42 #14H70 #35Q51 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2404.15200
openalex publication_date 2024/04/23 · openalex created_date 2024/04/26 · openalex updated_date 2026/07/28
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 ⟨ 2,5⟩. (In the theory of curve singularities, these are known as A4 double points.) We conjecture that these ideas will generalize to give similar M-lump solutions with M = (N(N+1))/(2) for N > 2 starting from rational curves with two singular points with semigroup ⟨ 2,2N+1⟩ (A2N double points). We also show a five-lump solution obtained from a curve with two cusps with semigroup ⟨ 3,4⟩. 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.