The Kadomtsev-Petviashvili (KP) equation is a nonlinear partial differential that is a fundamental one among all integrable systems. This talk aims to make an excursion into the study of the KP equation with a view from computational algebraic geometry. The emphasis will be on exploiting modern tools in symbolic, numerical and combinatorial algebraic geometry to investigate solutions of the differential equation.
In this talk, 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. After introducing solitons solutions, we compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces. This is joint work with Daniele Agostini, Yelena Mandelshtam and Bernd Sturmfels.