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At the Maxwell balance for a scalar reaction-diffusion front, let $q=3\alpha /4$, $\beta =\sqrt{3}\alpha /2$, and perturb the linear growth rate by $\epsilon K\delta (x+cT)$, $T=\epsilon t$. Projection onto the front's translation mode gives $X_T=-K{\mathrm{sech}}^2(\beta Z/2)$, where $Z=X+cT+{\beta }^{-1}\log q$. Hence $Z_T=c-K{\mathrm{sech}}^2(\beta Z/2)$. Finite locked positions exist for $0<c/K\le 1$. 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.

Set $f(A)={\mu }_{0}A+\alpha A^3-A^5$ and $V(A)={\mu }_0{A}^2/2+\alpha A^4/4-A^6/6$. Multiplication of the stationary reaction-diffusion equation by $A_x$ gives the first integral $A_x^2/2+V(A)=E$. A regular front with finite endpoint limits has $A_x\to 0$ there, and its nonzero endpoint must satisfy $f(C)=0$. Since $V(0)=0$, the first integral gives $E=0$ and $V(C)=0$. Write $q=C^2>0$. The two endpoint conditions areEliminating ${\mu }_0$ gives the Maxwell balance for a scalar reaction-diffusion front:This is a necessary condition for a stationary front, because the two spatially homogeneous states must have equal potential. It is also sufficient here. With these values, $-2V(A)=A^2(A^2-q{)}^2/3$. For the increasing positive front chooseIntegrating this logistic equation and writing $\beta =2q/\sqrt{3}=\sqrt{3}\alpha /2$ givesThe positive constant multiplying the exponential is arbitrary and can be absorbed into $X_0$; choosing it to be $q$ gives the printed normalization exactly. The profile tends to zero and $q$ at the two ends and its first-order equation implies $A_0^{\prime \prime }+f(A_0)=0$. Its negative is the front to $-\sqrt{q}$, with the same squared profile.

Differentiation of the stationary equation produces the translation modeThe operator $\mathcal{L}$ is self-adjoint. Twice applying integration by parts gives, for $R$ with vanishing boundary Wronskian,Bounded $R,R^{\prime }$ suffice, because the derivatives of the front decay exponentially. This proves the identity as a Fredholm solvability condition associated with translation invariance.