= Quartic impulse-response Laplace resolvent
{title2=$\widehat G(0,s)=[2\sqrt{s+a_3}\sqrt{2\sqrt{a_2}\sqrt{s+a_3}-a_1}]^{-1}$}
For the whole-line equation $A_t+a_1A_{xx}+a_2A_{xxxx}+a_3A=0$ with $\operatorname{Re}a_2>0$, the <dispersion relation> is $s=a_1k^2-a_2k^4-a_3$. The temporal <Laplace transform> of the impulse response at zero is the real-k integral of $[s+a_3-a_1k^2+a_2k^4]^{-1}/(2\pi)$. Factor the denominator as $a_2(k^2+\kappa_1^2)(k^2+\kappa_2^2)$ with $\operatorname{Re}\kappa_j>0$ for large $\operatorname{Re}s$. 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>.
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