“…Such an expansion is admittedly questionable in view of Pawula's theorem, but can be controlled when manipulated with care (Popescu and Lipan, 2015). Thus, the theorem of Pawula does not necessarily restrict the truncation of higher order terms, when we can formally obtain high-order perturbative equations (Kanasava, 2017) and nonvanishing higher-order coefficients have been observed in various systems (Anvari et al, 2016;Friedrich et al;, 2011;Prusseit and Lehnertz, 2007;Tutkun and Mydlarski, 2004;Kim et al, 2008;Petelczyc et al, 2009Petelczyc et al, , 2015. Therefore, though higher-order perturbative models might, in some cases, have negative values at some isolated times and positions, this does not invalidate the models derived in the study, which are valid only in zones where large concentration gradients are applied.…”