= Backward-sign Airy half-line global relation
{title2=$e^{ik^3t}\widehat q(k,t)=Q(k)+k^2F_0-ikF_1-F_2$}
For the <Airy equation> with opposite dispersive sign, $q_t-q_{xxx}=0$ on $x>0$, let $Q$ be the initial <Half-range Fourier transform> and $F_j(k,t)=\int_0^t e^{ik^3s}\partial_x^jq(0,s)\,ds$. Three <integrations by parts> give the displayed <global relation>. The spatial transform is <holomorphic> for $\operatorname{Im}k<0$, whereas the <finite-time spectral boundary transforms> are <entire functions>. The symmetry $k\mapsto\alpha k$, $\alpha^3=1$, preserves their time exponent.
// Target: analysis.bigb
Back to article page
