Bloch stability operator ($L(q)=e^{-iq\cdot x}L{e}^{iq\cdot x}$)

For a spatially periodic nonlinear pattern, a linear stability disturbance is written $e^{\lambda t+iq\cdot x}v(x)$ with $v$ periodic on the pattern cell. The conjugated operator $L(q)$ acts on that fixed cell and determines the sideband eigenvalues. Translation invariance supplies zero eigenfunctions at $q=0$, namely spatial derivatives of the base pattern. Their small-$q$ eigenvalues determine long-wavelength phase stability; other parts of the spectrum can give independent instabilities.

Suppose a Bloch stability operator has expansion $L(q)=L_0+iq{L}_1-q^2{L}_2+\cdots$, with a simple neutral eigenfunction $v_0$ and an adjoint neutral function $w$ normalized by $⟨w,v_0⟩=1$. Expand $v(q)=v_0+iq{v}_1+\cdots$ and $\lambda (q)=icq-D{q}^2+\cdots$. The Fredholm alternative gives $c=⟨w,L_1{v}_0⟩$ and the cell problem $L_0{v}_1=c{v}_0-L_1{v}_0$, $⟨w,v_1⟩=0$. Applying the adjoint solvability condition at second order gives the displayed phase-diffusion coefficient. Multiple translation phases or a conserved mean require a coupled matrix problem rather than this scalar formula.