Logistic crossover in a weakly nonlinear Euler equation ($H(X)=X-\log (1+X)$)

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

When the early growing solution is $A{x}^2$ and the nonlinear coefficient depends on $\epsilon y^{\prime }$, its crossover occurs at the displayed scale. If $A$ is order one this is $x=O({\epsilon }^{-1})$; if the initial slope suppresses the leading mode and $A=O(\epsilon )$, it is delayed to $x=O({\epsilon }^{-2})$. The first nonzero perturbative seed must therefore be computed before constructing the global leading approximation; setting the small parameter to zero can erase the mode that ultimately dominates.