Vladimir Matveev (Friedrich Schiller University Jena) – Finite-Gap and Soliton Solutions of BKM Systems

The BKM system is a recently  (2022) discovered multicomponent integrable system of partial differential equations. These multicomponent, nonlinear, dispersive systems display features characteristic of physically relevant models: covariance, infinitely many conservation laws, hidden symmetries, and the existence of compactly supported and quasi-periodic solutions. Its special cases include well-known equations such as the Korteweg–de Vries (KdV), coupled KdV, Harry Dym, coupled Harry Dym, Camassa–Holm, multicomponent Camassa–Holm, Dullin–Gottwald–Holm, and Kaup–Boussinesq equations.

I will start my presentation with an introduction and discussion of this system. The primary objective is to outline a methodology for constructing a series of solutions for the BKM system. The crux of the approach lies in reducing this system to a dispersionless integrable system, which is a minor generalisation of the so-called linearly degenerate systems. These infinite-dimensional integrable systems are closely linked to finite-dimensional integrable systems arising from the Stäckel construction; the finite-dimensional systems are quantum-integrable in the sense of Carter.

The solutions discussed for BKM systems are given in quadratures, in the sense that they are implicitly defined by primitive functions of explicitly given closed 1-forms. The set of solutions one can construct by this method is locally dense for certain BKM systems. I will also discuss numerical methods that enable us to visualise these solutions effectively. The results are joint with A. Bolsinov and A. Konyaev.