2025/07/10 by Kostya Druzhkov, Druzhkov, Kostya
Engineering · Mathematics · Physics and Astronomy · #Control and Stability of Dynamical Systems #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2507.08213
openalex publication_date 2025/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that, under suitable conditions, finite-dimensional systems describing invariant solutions of partial differential equations (PDEs) inherit local Hamiltonian operators through the mechanism of invariant reduction, which applies uniformly to point, contact, and higher symmetries. The inherited operators endow the reduced systems with Poisson bivectors that relate constants of motion to symmetries. Applying the same mechanism to invariant conservation laws, we further show that the induced Poisson brackets agree with those of the original systems, up to sign. The results are illustrated by two examples in which the inherited Poisson brackets and inherited constants of motion yield integrability of the reduced systems. The construction is independent of the choice of an ℓ-normal inclusion of a PDE system into jet spaces.