= Solution
Differentiate the <Airy ordinary differential equation> $y''=sy$ to obtain $y^{(3)}=y+sy'$. Combining part (a)'s <derivatives> yields
$$
u_t+u_{xxx}=[at^{a-1}+b^{3a}t^{4a}][y(s)+sy'(s)].
$$
To make this vanish for an arbitrary Airy profile, the time powers must agree: $a-1=4a$, giving $a=-1/3$. Their common <coefficient> must then vanish, giving $-1/3+b^{-1}=0$, hence $b=3$. Therefore
$$
\boxed{a=-\frac13,\qquad b=3,\qquad
u(x,t)=t^{-1/3}y\left(\frac{x}{(3t)^{1/3}}\right).}
$$
For $t>0$ the two terms cancel exactly:
$$
u_t=-\frac13t^{-4/3}(y+sy'),\qquad
u_{xxx}=\frac13t^{-4/3}(y+sy').
$$
This is an <Airy similarity solution of the linear dispersive equation>. The <Airy equation> here is the linear partial <differential equation> $u_t+u_{xxx}=0$, the reduced equation specified in the question; it is distinct from the profile's <Airy ordinary differential equation>. The construction works for every linear combination of its two independent profiles, including zero. No condition at $t=0$ is supplied, and this singular time is excluded. With real cube roots the final formula can also be used separately on $t<0$.
Back to article page
