= Moving-defect locking of a cubic-quintic front
At the <Maxwell balance for a scalar reaction-diffusion front>, let $q=3\alpha/4$, $\beta=\sqrt3\alpha/2$, and perturb the linear growth rate by $\varepsilon K\delta(x+cT)$, $T=\varepsilon t$. Projection onto the front's translation mode gives $X_T=-K\operatorname{sech}^2(\beta Z/2)$, where $Z=X+cT+\beta^{-1}\log q$. Hence $Z_T=c-K\operatorname{sech}^2(\beta Z/2)$. Finite locked positions exist for $0<c/K\le1$. For a strict inequality, the root with $KZ<0$ is stable and the other unstable; they meet in a <saddle-node bifurcation> at $c=K$. This is a leading-order <solvability condition>, not an exact finite-perturbation solution.
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