Computable Analysis of a Boundary-Value Problem for the Korteweg-de Vries Equation

The initial-boundary-value problem for the Korteweg-de Vries (KdV) equation:

$u_{t}+u_x+uu_{x}+u_{xxx}=0, \quad t, x\geq 0,$

$u(x, 0)=\varphi (x),\quad u(0,t)=h(t), \qquad\varphi(0)=h(0),$

defines a nonlinear continous map from the space where the auxiliary data are drawn to the space of solutions. By making use of modern methods for the study of nonlinear dispersive equations, it is shown that the solution map\(H^{3m-1}({\Bbb R}^+)\times H^{m}(0,T)\to C(0,T];H^{3m-1}({\Bbb R}^+))\) is Turing computable for any integer m ≥ 2 and computable real number T > 0. This result provides yet another affirmative answer to the open question raised by Pour-El and Richards PER]: Is the solution operater of the KdV equation computable?