Computing the solution of the Korteweg-de Vries equation with arbitrary precision on Turing machines

In this paper we answer an open question raised by Pour-El and Richards: Is the solution operator of the Korteweg-de Vries (KdV) equation computable? The initial value problem of the KdV equation posed on the real line R: ut+uux+uxxx=0,t,x∈R,u(x,0)=ϕ(x)defines a nonlinear map KR from the space Hs(R) to the space C(R;Hs(R)) for real numbers s⩾0. We prove that for any integer s⩾3, the map KR:Hs(R)→C(R;Hs(R)) is Turing computable.