Organizers
Vladimir Matveev, Friedrich Schiller University, Germany
Youjin Zhang (张友金), Tsinghua University, China
Zhijun Qiao (乔志军), University of Texas Rio Grande Valley, USA
Andrey Konyaev, Lomonosov Moscow State University, Russia
Abstract
Integrable systems and solitons appear in many fields, including fluid mechanics, plasma physics, optics, and differential geometry, and have undergone tremendous developments over the past few decades. They are important components of applied mathematics and nonlinear sciences, with profound relations to nonlinear water wave equations, Lie algebras, PDE analysis, symplectic geometry, and other branches of mathematics, as well as significant applications in nonlinear optics, fluid dynamics, theoretical mechanics, theoretical physics, mathematical physics, and many other natural and social sciences.
The conference will bring together experts from fields including Riemannian geometry, Frobenius manifolds, the theory of integrable systems and solitons, dynamical systems, algebraic geometry, symplectic geometry, PDEs, and mathematical physics, to exchange topics and suggestions and to share various approaches for studying different problems in integrable systems and solitons, applying these ideas and methods in their respective research projects.
One of the foci of the conference will be BKM systems, a recently discovered (2022) family of integrable partial differential equation systems arising from Nijenhuis geometry, marking a new and challenging development at the intersection of differential geometry and mathematical physics. 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. Special cases recover several classical integrable models of central importance in physics and hydrodynamics, such as the KdV
and Camassa–Holm equations, as well as the Kaup–Boussinesq and Ito systems.
Our goal is to develop methods for universal treatment of the entire BKM family. While individual cases such as the KdV equation have been extensively studied, central challenges—including the construction of exact solutions, the inverse scattering method, the finite-gap approximation of arbitrary solutions, and ultimately quantization—remain unresolved for most members of the family. By addressing these problems in a unified framework, the project aims to achieve ambitious and innovative advances in integrable systems theory.
The novelty of our approach lies in combining two complementary perspectives. On the one hand, we exploit modern differential geometry, in particular geometrically invariant structures provided by Nijenhuis geometry. On the other, we draw on analytical and algebraic techniques motivated by physics and rooted in seminal works by Gelfand–Dikii, Kruskal–Zabusky, Novikov, Dubrovin, Krichever, Veselov, Moser, Van Moerbeke, McKean, Trubowitz, Faddeev, Its, Zhang, and others. This synergy yields methodological tools that extend far beyond what has been possible for isolated models.
