Computing the solution of the Korteweg-de Vries equation with arbitrary precision on Turing machines
The Korteweg-de Vries (KdV) equation \(u_t + u u_t + u_{ttt} =0\) (\(t, x \in \mathbb{R}\)) with the initial condition \(u(x, 0)= \varphi(x)\) defines an operator which maps a function \(\varphi\) to the solution of the KdV equation. In Math. Log. Q. 47, No. 1, 93--110 (2001; Zbl 0970.03047), \textit{W. Gay, B.-Y. Zhang} and the second author show that this operator preserves the computability, i.e., it maps a computable initial function to the computable solution. This paper extends this result to a more uniform version, namely, this operator is actually a computable function from the Sobolev space \(H^s(\mathbb{R})\) with \(s\geq 3\) to the space \(C(\mathbb{R}, H^s(\mathbb{R}))\) of continuous function. This answers an open question of \textit{M. B. Pour-El} and \textit{J. I. Richards} [Computability in analysis and physics. Berlin etc.: Springer (1989; Zbl 0678.03027)].
scientific article; zbMATH DE number 2153191
scientific article