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.
I start with the definition of cluster integrable systems a la Goncharov and Kenyon, defined by convex Newton polygons, up to the action of $SA(2,\mathbb{Z})$. There are several arguments requiring that to complete the picture, this class should be extended by their Hamiltonian reductions, which can be performed preserving the structure of cluster variety.
In this talk, we consider the focusing Nonlinear Schrödinger (NLS) equation and its multisoliton solution when the number of solitons grows to infinity. We discover configurations of multi-soliton solutions that exhibit the formation of a soliton gas condensate. Specifically, we show that, when the associated discrete eigenvalues accumulate on two bounded horizontal segments in the complex plane and the associated norming constants are bounded away from zero, the solution to the focusing NLS equation is described by a rapidly oscillatory elliptic wave with constant velocity, on compact subsets of the $(x,t)$ domain.
On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson structure, the ``r-matrix structure’’. It is an essential structure underlying the Hamiltonian dynamics of the vast majority of integrable systems, isospectral and isomonodromic evolution equations. The known r-matrices depend on parameter in rational way (trig/elliptic, respectively) and hence we think of them on the Riemann sphere (cylinder/torus).
In a relatively abstract Hamiltonian framework the isospectral evolution equations are generalized to higher genus Riemann surfaces as the “Hitchin systems”, an evolutionary integrable system on the moduli space of vector bundles. On the isomonodromic side main progress is attributable to Krichever who used a quite explicit coordinatization of vector bundles on Riemann surfaces that we can call “Tyurin parametrization”.
In this talk I report on the fully explicit generalization of the r-matrix structure to an arbitrary genus Riemann surface merging the Tyurin-Krichever approach with the general framework of Hitchin’s. The key tool is a (fully explicit) matrix-valued kernel that plays crucial role also in setting up integral equations in related area of the "non-abelian steepest descent” method.
We briefly review main examples of elliptic integrable systems including Calogero-Moser system, its spin generalization, integrable tops and Gaudin type models. Then we describe their field generalizations through the Zakharov-Shabat U-V pairs and ultralocal or non-ultralocal classical r-matrix structure of Maillet type. For example, the elliptic top is extended to the Landau-Lifshitz model. These type models are also described using R-matrices satisfying the associative Yang-Baxter equation. This allows to include into consideration a wide class of trigonometric and rational models. Next, we procced to relativistic systems including the Ruijsenaars-Schneider model, relativistic tops, classical spin chains and classical Ruijsenaars chains. Their field versions are described by the semi-discrete Zakharov-Shabat equations and quadratic r-matrix structures.
