Hi,
The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the dispersion relation was $-\alpha^3$ and then to have a Fourier expansion of the form $$u(x,t)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i(kx+k^3t)}\int_{-\infty}^\infty e^{-iky}dydk $$ then by playing around with $k=\alpha/(3t)^{1/3}$ I get the following terms
\begin{align} u(x,t)&=\frac{1}{2\pi}\int_{-\infty}^\infty \frac{u_0(y)}{(3t)^{1/3}}dy\int_{-\infty}^\infty e^{i(\alpha\frac{x-y}{(3t)^{1/3}}+\frac{\alpha^3}{3})}d\alpha\\ &=\frac{1}{2\pi}\int_{-\infty}^\infty \frac{u_0(y)}{(3t)^{1/3}}\Bigg[\int_{-\infty}^\infty e^{i(\alpha\frac{x}{(3t)^{1/3}}+\frac{\alpha^3}{3})}d\alpha-\frac{y}{(3t)^{1/3}}\int_{-\infty}^\infty\alpha e^{i(\alpha\frac{x}{(3t)^{1/3}}+\frac{\alpha^3}{3})}d\alpha\Bigg]dy+\dots \end{align} where, to arrive at the second expression, we need to use taylor expansion for $exp(-iky)$. Now, I'm trying to find an asymptotic expansion in the area $x/t^{1/3}\approx O(1)$, for $x>0$ and $x<0$ separately... Drazin's book states that $x<0$ should be "a slowly decaying wave train" and $0<x$ should be "a steeply rising wave front" But i don't know how to arrive at that.
Could anyone lend a hand with the task?
edit:
After awhile trying to find the asymptotics using the asymptotics or $Ai(x)$ I've come to the conclusion that it is not the way since they are based on $\frac{x}{(3t)^{1/3}}\to\infty$ and are so useless to my effort. thus focusing on \begin{align} u(x,t)&=\int_{-\infty}^\infty \frac{u_0(y)}{(3t)^{1/3}}dy\cdot Ai(\frac{x}{(3t)^{1/3}})-\int_{-\infty}^\infty y\frac{u_0(y)}{(3t)^{2/3}}dy\cdot Ai'(\frac{x}{(3t)^{1/3}})+\dots \end{align} I've decided to look for the asymptotics for fixed $v=\frac{x}{(3t)^{1/3}}=o(1)$ which leads to: \begin{align} u(v(3t)^{1/3},t)&=\frac{v}{x}\int_{-\infty}^\infty u_0(y)dy\cdot Ai(v)-\frac{v^2}{x^2}\int_{-\infty}^\infty y\cdot u_0(y)dy\cdot Ai'(v)+\dots \end{align}
does anyone have an idea how to continue?