= Logistic crossover in a weakly nonlinear Euler equation
{title2=$H(X)=X-\log(1+X)$}
A growing linear mode can make a formally small <derivative> nonlinearity order one at large distance. After the crossover scaling, the equation $XH''-H'+(H')^2=0$ reduces to a <logistic differential equation> in $\log X$ for $H'$. Matching $H\sim X^2/2$ at small $X$ gives $H'=X/(1+X)$ and the displayed primitive with $H(0)=0$. It changes from quadratic growth to linear growth without a finite turning singularity.
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