On the Algorithmic Recovering of Coefficients in Linearizable Differential Equations
Dmitry A. Lyakhov, Dominik L. Michels
TL;DR
The paper addresses recovering coefficients in scalar nonlinear ODEs that admit exact linearization via an invertible transformation. It proposes an algorithmic framework based on Lie symmetry analysis, computing the finite-dimensional symmetry algebra $L$ and its derived algebra $\mathcal{D}$ to certify linearizability, while a linearizing map is obtained by solving a Bluman-Kumei system; in the constant-coefficient subcase, the approach leverages the characteristic polynomial data. A dedicated coefficient-recovery algorithm is developed for the case $m=n+2$, which complements existing tools such as LinearizationTestII and MapDE, and ties spectral properties of the symmetry algebra to the recovered linear form. The work provides a principled, algebraic route to certify linearizability and reconstruct linearizing mappings for quasi-linear ODEs with rational right-hand sides, enabling more automated linearization within symbolic computation workflows.
Abstract
We investigate the problem of recovering coefficients in scalar nonlinear ordinary differential equations that can be exactly linearized. This contribution builds upon prior work by Lyakhov, Gerdt, and Michels, which focused on obtaining a linearizability certificate through point transformations. Our focus is on quasi-linear equations, specifically those solved for the highest derivative with a rational dependence on the variables involved. Our novel algorithm for coefficient recovery relies on basic operations on Lie algebras, such as computing the derived algebra and the dimension of the symmetry algebra. This algorithmic approach is efficient, although finding the linearization transformation necessitates computing at least one solution of the corresponding Bluman-Kumei equation system.