KP Solitons from Tropical Limits

Abstract

We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve.

Nonlinear Algebra and Applications

“…On the other hand, thinking of the quasi periodic solutions arise via the Riemann theta function, one may also study degenerations of the Riemann theta function for establishing such a relation. In fact, if one considers tropical degenerations of algebraic curves, the Riemann theta function happens to be a finite sum of exponentials [10, Theorem 4], [13,Theorem 3]. The Hirota variety parametrizes all the tau functions arising from such a sum [13].…”

“…In fact, if one considers tropical degenerations of algebraic curves, the Riemann theta function happens to be a finite sum of exponentials [10, Theorem 4], [13,Theorem 3]. The Hirota variety parametrizes all the tau functions arising from such a sum [13]. It would be worthwhile to discover more about possible connections of the Hirota variety with the soliton solutions [13,Example 11], also its potential link to the Dubrovin variety.…”

“…The Hirota variety parametrizes all the tau functions arising from such a sum [13]. It would be worthwhile to discover more about possible connections of the Hirota variety with the soliton solutions [13,Example 11], also its potential link to the Dubrovin variety. In dimension 3, the Riemann theta function degenerations recover the so-called theta surfaces [10], classically the double translation surfaces [150].…”