Finite-Dimensional Reductions and Finite-Gap-Type Solutions of Multicomponent Integrable PDEs
Bolsinov A.V. Konyaev A.Y. Matveev V.S.
August 2025John Wiley and Sons Inc
Studies in Applied Mathematics
2025#155Issue 2
The main object of the paper is a recently discovered family of multicomponent integrable systems of partial differential equations, whose particular cases include many well-known equations such as the Korteweg–de Vries, coupled KdV, Harry Dym, coupled Harry Dym, Camassa–Holm, multicomponent Camassa–Holm, Dullin–Gottwald–Holm, and Kaup–Boussinesq equations. We suggest a methodology for constructing a series of solutions for all systems in the family. The crux of the approach lies in reducing this system to a dispersionless integrable system which is a special case of linearly degenerate quasilinear systems actively explored since the 1990s and recently studied in the framework of Nijenhuis geometry. These infinite-dimensional integrable systems are closely connected to certain explicit finite-dimensional integrable systems. We provide a link between solutions of our multicomponent PDE systems and solutions of this finite-dimensional system, and use it to construct animations of multicomponent analogous of soliton and cnoidal solutions.
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School of Mathematics, Loughborough University, Loughborough, United Kingdom
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Faculty of Mechanics and Mathematics and Center for Fundamental and Applied Mathematics, Moscow State University, Moscow, Russian Federation
Institut für Mathematik, Friedrich Schiller Universität Jena, Jena, Germany
School of Mathematics
Institute of Mathematics and Mathematical Modeling
Faculty of Mechanics and Mathematics and Center for Fundamental and Applied Mathematics
Institut für Mathematik
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026