On the optimal systems of subalgebras for the equations of hydrodynamic stability analysis of smooth shear flows and their group-invariant solutions
Their claim that they have derived "a generalized approach" to stability analysis for unbounded shear flow that "unites both – the nonmodal (Kelvin mode) and the modal – approaches" for which "the class of solutions is much wider than the well-known Kelvin and modal solutions" is not correct and therefore misleading.
First, their self-defined "modal approach" and "Kelvin mode approach" are complementary approaches, namely in being a pure temporal and a pure spatial approach respectively, which exclude each other and which therefore cannot be "united" into a single universal approach — this flaw is discussed in detail in Sec.4.2 in https://arxiv.org/abs/1712.03105.
Second, when choosing the correct reference frame, their "new generalized approach" becomes redundant in that it fully reduces either to their purely temporal "modal approach" or to their purely spatial "Kelvin mode approach". Hence, there are no physically "new" or "more general" modes which go beyond the ones already established. This problem is already encountered in two of their earlier publications: https://doi.org/10.1063/1.4823508 and https://doi.org/10.1063/1.4934726. For a detailed discussion, please see https://arxiv.org/abs/1712.03105.
The message that needs to be conveyed here is that when Lie-group symmetry analysis is applied to linear stability analysis, one has to be aware of the relative-motion aspect and the possibility to generate redundant modes even if they are group-theoretically inequivalent.
For example, the unnecessarily complicated single mode put forward in Sec.[V.C1] is nothing else than the simple spatial 2D Fourier mode when only reformulated in the correct reference frame. Since the authors were not aware of this connection, it is not surprising that they provoked the circumstance to make mistakes, as all their derivations and calculations grew overly complicated when moving forward in this unnecessarily overcomplicated representation. For example, in Sec.[V.C3] it becomes clear that the authors were unable to identify from this muddle the correct integration variables in order to yield from the set of single modes the correct full 2D spatial field and its complementary inverse, the full 2D wavenumber field. The correct full 2D wavenumber field (for $t=0$) is therefore not given through their Eq.(44), but, as given in https://arxiv.org/abs/1712.03105 by the mathematically completely different expression Eq.(4.5) (for $t=0$) — this latter result Eq.(4.5) even gives the correct general result for $t\neq 0$, which the authors avoided to do with the excuse to maintain "simplicity".
Finally to note here is that throughout their study the authors coin the "Kelvin mode approach" also as a "non-modal" approach, which is confusing and misleading, since the "Kelvin mode approach", as the name already says, is indeed a modal approach. The modes are given through their Eq.(21), which in the appropriate (optimal) frame would just reduce to the spatial Fourier modes — for more details, see https://arxiv.org/abs/1712.03105. Hence, this "Kelvin mode approach" is not to be confused with the proper (and different) non-modal approach as originally defined in the community of hydrodynamic stability (see e.g. the book by Schmid & Henningson: Stability and Transition in Shear Flows, 2001).