Theory of Computing Systems

, Volume 41, Issue 1, pp 155–175 | Cite as

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

  • Ning Zhong
Article

Abstract

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?

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Copyright information

© Springer 2007

Authors and Affiliations

  • Ning Zhong
    • 1
  1. 1.Department of Mathematical Sciences, University of CincinnatiCincinnati, OH 45221USA

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