Fifth R. E. Moore Prize

  • R. E. Moore Prize (2014)
  • I have been awarded the 5th R. E. Moore Prize for Applications of Interval Analysis.
  • The R. E. Moore Prize was established in 2002 by the Editorial Board of Reliable Computing, an International Journal devoted to reliable mathematical computations based on finite representations and guaranteed accuracy.
  • The prize is awarded for our paper
      Kenta Kobayashi, Computer-Assisted Uniqueness Proof for Stokes' Wave of Extreme Form, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, Vol.10 (2013), pp.54-67
    which is the summarized version of the paper
      K. Kobayashi, On the global uniqueness of Stokes' wave of extreme form, IMA Journal of Applied Mathematics, Vol. 75(5) (2010), 647-675.
  • Details
  • We have proved existence of positive solutions of Nekrasov's equation and Stokes' waves of extreme form using validated computation.
  • The gravity and the surface tension have much influence on the form of water waves. Assuming that the flow is infinitely deep, the gravitational acceleration is a unique external force of the system and the wave profile is stationary, we obtain Nekrasov's equation. In particular, a positive solution of Nekrasov's equation corresponds to a water wave which has just one peak and one trough per period.
  • Stokes' wave of extreme form has a sharp crest and is considered to be the limit of the positive solution of Nekrasov's equation with respect to parameters such as gravity, wave length, and wave velocity. Proofs for existence of the positive solutions of Nekrasov's equation and Stokes' wave of extreme form appear in
      G. Keady & J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Camb. Phil. Soc., Vol. 83 (1978), 137-157,
    and
      J. F. Toland, On the existence of a wave of greatest height and Stokes's conjecture, Proc. R. Soc. Lond. A, Vol. 363 (1978), 469-485,
    respectively. However, global uniqueness of these waves had not been proved for a long time. We suppose it is almost 30 years during which a lot of efforts were devoted in order to solve the problem.
  • We proved the global uniqueness of the positive solution of Nekrasov's equation for a certain extent of parameters in 2004:
      K. Kobayashi, Numerical verification of the global uniqueness of a positive solution for Nekrasov's equation, Japan Journal of Industrial and Applied Mathematics, Vol. 21(2)(2004), 181-218
    and finally we proved the global uniqueness of Stokes' wave of extreme form in 2010:
      K. Kobayashi, On the global uniqueness of Stokes' wave of extreme form, IMA Journal of Applied Mathematics, Vol. 75(5) (2010), 647-675.
  • These researches brought an epoch-making advance on a historical problem of water wave which has been actively investigated worldwide.
  • Stokes' conjecture
  • The uniqueness of Stokes' wave concerns the second Stokes' conjecture which appeared in Stokes' paper published in end of the 19th century:
      G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, Math. Phys. Papers, Vol. 1(1880), 225-228.
    The first Stokes' conjecture supposed that Stokes' wave has sharp crests of included angle 120 degree, and this is proved in
      C. J. Amick, L. E. Fraenkel, & J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., Vol. 148 (1982), 193-214.
    The second conjecture supposed that the profile of Stokes' wave between two consecutive crests should be downward convex, which has been a longtime open problem for 130 years.
  • Existence of Stokes' wave of extreme form which has the profile of downward convex was proved by J. F. Toland and P. I. Plotnikov in 2004:
      J. F. Toland, & P. I. Plotnikov, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal., Vol. 171 (2004), 349-416.
    Therefore, the complete settlement of the second Stokes' conjecture was brought by our result in 2010 which proves the global uniqueness of Stokes' wave of extreme form.
  • Global uniqueness
  • Validated computation is used for the proof of the global uniqueness. We changed the concerned equation into global contraction mapping by an ingenious transformation, and verified the contractiveness by validated computation, which leads to the proof of global uniqueness.
  • In the area of application of validated computation, there are a lot of results obtained to prove local uniqueness of solutions of functional equations. On the other hand, only a few results are obtained for proofs of global uniqueness. Our results are extremely rare examples of validated computation for proving global uniqueness, which should be really interesting and highly important applications of interval arithmetic.