Reducibility of zero curvature equations. (English) Zbl 1058.34049

A pair \(( \Theta _{1},\Theta _{2}) \) of smooth complex \(n\times n\)-matrix functions in the variables \(( s,t) \in \mathbb{R}^{2}\) is called a compatible pair if there exists a smooth complex function \(G( s,t) \) satisfying \[ \frac{\partial G}{\partial s}=\Theta _{1}G, \quad \frac{\partial G}{\partial t}=\Theta _{2}G, \] with the initial condition \(G( 0,0) =I_{n}\). The commutativity of the second mixed derivatives \(\frac{\partial ^{2}G}{\partial s\partial t}=\frac{\partial ^{2}G}{\partial t\partial s}\) implies \(\frac{\partial \Theta _{1}}{\partial t}-\frac{\partial \Theta _{2}}{\partial s}+[ \Theta _{1},\Theta _{2}] =0\) which is called the zero-curvature equation, because the pair \(( \Theta _{1},\Theta _{2}) \) yields a linear connection \(\mathbb{R}^{2}\times \mathbb{C}^{n}\to \mathbb{R}^{2}\) with the connection 1-form \(\Omega =-( \Theta _{1}ds+\Theta _{2}dt) \) having the curvature \(d\Omega +\Omega \wedge \Omega =0\).
For this type of systems, a natural concept of reducibility is introduced as the existence of a global “gauge” transformation reducing the initial system to another one with constant coefficients. The main result is a Floquet representation for such equations. As applications, the reducibility problems for quasiperiodic 2-dimensional linear systems and for fiberwise linear dynamical systems on trivial vector bundles are discussed.


34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A26 Geometric methods in ordinary differential equations
34A30 Linear ordinary differential equations and systems
37C55 Periodic and quasi-periodic flows and diffeomorphisms
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