  This study introduces a high-order perturbation methodology to categorize two
primary solution types within duality-invariant nonlinear electrodynamic
theories, adhering to the differential self-duality criterion. The first
solution type aligns with irrelevant stress tensor flows, resembling $T\bar{T}$
dynamics, and the second involves a blend of irrelevant $T\bar{T}$-like and
marginal root-$T\bar{T}$-like deformations. Our approach facilitates the
investigation of diverse duality-invariant nonlinear electrodynamics theories
and their stress tensor flows and confirms the commutativity of flows initiated
by irrelevant and marginal operators.
