Description
We study approximating underlying mathematical structures via point cloud data within the framework of integrable systems, with a particular emphasis on the Kadomtsev-Petviashvili (KP) equation. This is a pivotal element in the theory of integrable systems, and models nonlinear wave interactions. Our objective is to determine the finite-genus KP solution parameters that optimally correspond to a prescribed dataset of discrete point values. The methodology employs foundational principles from Fourier analysis alongside standard optimization techniques. A beautiful mathematical challenge we run into is the Schottky problem, which asks for characterization of Jacobians of algebraic curves among abelian varieties. This is joint with Daniele Agostini, Bernard Deconinck, Charles Wang.
