Quartic impulse-response Laplace resolvent ($\hat{G}(0,s)=[2\sqrt{s+a_3}\sqrt{2\sqrt{a_2}\sqrt{s+a_3}-a_1}{]}^{-1}$)

For the whole-line equation $A_t+a_1{A}_{xx}+a_2{A}_{xxxx}+a_{3}A=0$ with $\mathrm{Re}{a}_2>0$, the dispersion relation is $s=a_1{k}^2-a_2{k}^4-a_3$. The temporal Laplace transform of the impulse response at zero is the real-k integral of $[s+a_3-a_1{k}^2+a_2{k}^4{]}^{-1}/(2\pi )$. Factor the denominator as $a_2(k^2+{\kappa }_1^2)(k^2+{\kappa }_2^2)$ with $\mathrm{Re}{\kappa }_j>0$ for large $\mathrm{Re}s$. Partial fractions give the displayed resolvent. Square-root sheets are chosen by continuation from this causal large-s region; choosing roots independently can introduce false spatial pinch points.

Put $z=a_1/\sqrt{a_2}$ using the root with positive real part. The additional resolvent singularity at $s+a_3=z^2/4$ lies on the physical square-root sheet only when $\mathrm{Re}z>0$; otherwise the apparent saddle is on the other sheet. The singularity at $s=-a_3$ remains, including a quartic degeneracy when $a_1=0$. The rightmost physical singularity determines the exponential fixed-position growth rate. Compare it with the temporal maximum $-\mathrm{Re}{a}_3+\max (\mathrm{Re}{a}_1,0{)}^2/(4\mathrm{Re}{a}_2)$ to distinguish absolute wave-packet instability from convective wave-packet instability. Equality cases require their algebraic prefactors.