The discrete KP and KdV equations over finite fields

TL;DR

This paper develops an algebro-geometric method to construct solutions for the discrete KP and KdV equations over finite fields, enabling multisoliton solutions from vacuum wave functions on algebraic curves.

Contribution

It introduces a novel algebro-geometric approach for finite field solutions of discrete integrable equations, including explicit multisoliton formulas.

Findings

Constructed multisoliton solutions over finite fields.

Reduced discrete KP to discrete KdV over finite fields.

Provided explicit formulas from vacuum wave functions.

Abstract

We propose the algebro-geometric mothod of construction of solutions of the discrete KP equation over a finite field. We also perform the corresponding reduction to the finite field version of the discrete KdV equation. We write down formulas which allow to construct multisoliton solutions of the equations starting from vacuum wave functions on arbitrary non-singular curve.