Jet-Density of Finite-Gap Solutions for Classes of BKM Systems

TL;DR

The paper demonstrates that finite-gap solutions can approximate jets of initial data for certain BKM PDE systems, including KdV, Kaup–Boussinesq, and Camassa–Holm, with varying degrees of surjectivity.

Contribution

It introduces an algebraic finite-reduction map that connects solutions of Stäckel systems to BKM PDEs, establishing jet-surjectivity results for key classes.

Findings

Finite-gap solutions approximate jets of initial data for BKM systems.

Full jet-surjectivity achieved for KdV and Kaup–Boussinesq classes.

Partial jet-surjectivity established for Camassa–Holm class over real and complex domains.

Abstract

We show that jets of initial data can be approximated up to arbitrary order by finite-gap solutions for classes of so-called BKM systems of PDEs introduced by Bolsinov--Konyaev--Matveev, which include classical PDEs such as KdV, Kaup--Boussinesq and Camassa--Holm. Finite-gap solutions are obtained via a finite-reduction map, defined algebraically, which sends solutions of a St\"ackel system to solutions of the BKM PDE. For the classes containing KdV and Kaup--Boussinesq we obtain full jet-surjectivity via a triangular structure, whereas for the class containing Camassa--Holm we establish jet-surjectivity on an open set of initial data over $\mathbb{R}$ and a Zariski-open (dense) set over $\mathbb{C}$.