At his 60th birthday conference in 2005, Percy Deift was asked to present a list of unsolved problems. This list was updated ten years later, on his 70th birthday conference. As the number one unsolved problem in both lists we still have the following conjecture.
Problem 1.1 (KdV with almost periodic initial data). Consider the Korteweg–de Vries (KdV) equation
$$u_t +uu_x +u_{xxx} =0 $$
with initial data
$$u(x,t=0)=q(x),\quad x\in\mathbb{R}.$$
In the 1970’s, McKean and Trubowitz proved the remarkable result that if the initial data $q(x)$ is periodic, $q(x + T ) = q(x)$ for some $T > 0$, then the solution $u(x, t)$ is almost periodic in time. This result leads to the following natural conjecture: The same is true if $q(x)$ is almost periodic, i.e., if the initial data is almost periodic in space, the solution evolves almost periodically in time.
Zoom Link:
https://msu.zoom.us/j/94297154840
Passcode: the same as the last time