= Airy half-line solution with prescribed boundary derivative
{c}
For $u_t+u_{xxx}=0$ on $x>0$, define $V(x,p)=\int_0^\infty R_p(x-y)u_0(y)dy$ and let $G$ be the time <Laplace transform> of $u_x(0,t)$. The transformed solution with spatial decay is
$$
U(x,p)=V(x,p)+\frac{e^{-p^{1/3}x}}{p^{1/3}}\left[V_x(0,p)-G(p)\right].
$$
Only one <characteristic root> has negative <real part>, so one scalar boundary derivative fixes the remaining homogeneous mode. The <Bromwich inversion formula> supplies an <integral representation> involving only the initial and boundary data. For general <linear differential equations> on a half-line, the number and form of the necessary boundary data depend on the decaying spatial roots; https://arxiv.org/abs/1409.2083[Fokas and Wang] study the corresponding boundary maps for linear dispersive equations.
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