| Abstract: |
| The Kadomtsev-Petviashvili (KP) equation describes 3-dimensional waves in shallow water with weak dependence on the second spatial variable. Solutions of the KP equation with finitely many interacting phases can be expressed in terms of the Riemann theta function, and complex algebraic curves can be used to determine the parameters (e.g. wave vectors) therein. Particularly, a KP solution corresponding to a curve on a genus-$g$ surface will have $g$ phases present. We develop a new framework for the recovery problem: given a KP solution as a set of data values, we determine the corresponding genus and parameters. Our framework is rooted in Fourier analysis alongside a non-linear optimization step, which we recast as two linear least-squares problems. Whereas previous works were limited to recovery of genus 2 solutions, our framework is applicable to higher genus. |
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