Given $u_t=u_{xxx}$, consider initial conditions consisting of two wave packets. The first wave starts at $x_1=0$ and has wave number $k_1=2$. The second wave starts at $x_2=10$ and has wave number $k_2=1$. When and where will their paths cross?
Attempted Solution
Assuming that the solutions are of the form $u(x,t)=e^{i(kx-\omega t)} $, we know that the dispersion relationship is $\omega =k^3$. Then, we can say,
$u(x,t) = e^{i(kx-\omega t)} = e^{i(kx-k^3t)}$
Then, plugging the values $x_1, k_1, x_2, k_2$ we can find equations for the two waves, $u_1, u_2$:
$u_1(x,t) = e^{i(2*0-8t)}$
$u_2(x,t) = e^{i(1*10-t)}$
Setting these two equations equal to solve for $t$, we find
$t=\frac{2}{7}(\pi n - 5)$ where $n\in\mathbb{Z}$
However this implies that there are negative values of $t$ that exist as solutions? Can someone point me in the right direction?