Dynamics of Partial Differential Equations

Volume 20 (2023)

Number 1

Applications of Nijenhuis geometry IV: Multicomponent KdV and Camassa–Holm equations

Pages: 73 – 98

DOI: https://dx.doi.org/10.4310/DPDE.2023.v20.n1.a4


Alexey V. Bolsinov (Department of Mathematical Sciences, Loughborough University, Loughborough, United Kingdom)

Andrey Yu. Konyaev (Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia; and Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia)

Vladimir S. Matveev (Institut für Mathematik, Friedrich Schiller Universität Jena, Germany)


We construct a new series of multi-component integrable PDE systems that contains as particular examples (with appropriately chosen parameters) and generalises many famous integrable systems including KdV, coupled KdV [1], Harry Dym, coupled Harry Dym [2], Camassa–Holm, multicomponent Camassa–Holm [14], Dullin–Gottwald–Holm and Kaup–Boussinesq systems. The series also contains integrable systems with no low-component analogues.


multicomponent integrable PDE systems, Korteweg–de Vries equation, Camassa–Holm equation, Harry Dym equation, Nijenhuis operator, evolutionary flow, conservation laws and symmetries

2010 Mathematics Subject Classification

Primary 37K10, 37K25, 37-xx, 53B50. Secondary 53A55, 53B20, 53D17.

The research of V.M. was supported by DFG grant MA 2565/7.

Received 16 November 2023

Published 23 December 2022