Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 18

Put $s=b^a{t}^{a}x$ and write $y^{\prime }$ for differentiation with respect to $s$. Work on a real branch where the powers are defined and differentiable; $b>0$, $t>0$ suffices for arbitrary real $a$, and the parameter choice below has $b>0$. At fixed $x$, $\partial s/\partial t=as/t$. The product rule and chain rule therefore give

At fixed $t$, the argument is linear in $x$, with $\partial s/\partial x=b^a{t}^a$. Each differentiation therefore contributes this same constant factor, while the outside factor $t^a$ remains unchanged. ThusHere $y^{(3)}$ denotes the third derivative of the single-variable profile, not a partial derivative of $u$.

Differentiate the Airy ordinary differential equation $y^{\prime \prime }=sy$ to obtain $y^{(3)}=y+s{y}^{\prime }$. Combining part (a)'s derivatives yieldsTo 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 $b=3$. ThereforeFor $t>0$ the two terms cancel exactly: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 $t<0$.