Koopman operator for Burgers's equation

We consider the flow of Burgers' equation on an open set (small) functions in $L^2([0,1])$. derive explicitly Koopman decomposition flow. identify frequencies and coefficients this as eigenvalues eigenfunctionals operator. prove convergence for $t>0$ small Cauchy data, up to $t=0$ regular data. The $t=0$} leads a `completeness' property basis modes. construct all modes eigenfunctionals, including eigenspaces involved geometric multiplicity. This goes beyond summation formulas provided by (Page & Kerswell, 2018), where only one term per eigenvalue was given. A numeric illustration is given compared Dynamic Mode Decomposition (DMD).