= Airy similarity solution of the linear dispersive equation
{c}
{title2=$u(x,t)=t^{-1/3}y(x/(3t)^{1/3})$}
Let $y$ solve the <Airy ordinary differential equation> $y''(s)=sy(s)$. For $t>0$, set $s=x/(3t)^{1/3}$ and define the displayed <similarity solution>. Differentiating the profile equation gives $y'''=y+sy'$, so the <chain rule> yields
$$
u_t=-\frac13t^{-4/3}(y+sy'),\qquad
u_{xxx}=\frac13t^{-4/3}y'''=\frac13t^{-4/3}(y+sy').
$$
Their sum is zero, proving the <Airy equation> $u_t+u_{xxx}=0$. Every real linear combination of the two standard <Airy functions> gives such a solution. This is a solution of the linear dispersive equation; the nonlinear <Korteweg-De Vries equation> includes an extra term and does not follow from the same arbitrary profile construction.
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