Korteweg-de Vries Institute for Mathematics

About the KdVI

D.J. Korteweg and G. de Vries

Korteweg

Life of D. J. Korteweg

In his memorial we read that Diederik Johannes Korteweg was born March 31, 1848 in the town of 's Hertogenbosch in the South of the Netherlands, where his father was a judge. This time may be seen as the dawn of the golden age of Dutch science, see the book by Levelt Sengers about Van der Waals and the Dutch School for more information on this period. It was marked by outstanding and influential investigations of, among others, J.D. van der Waals, H. de Vries, J.C. Kapteyn, J.H. van 't Hoff, H.A. Lorentz, H. Kamerlingh Onnes. One of the prominent scientists, who contributed so much to the cultural life in the Netherlands, was D.J. Korteweg.

Korteweg started his academic studies at the so called "Polytechnic School", now the Technical University of Delft. Because his disposition for mathematics was stronger than that for technical sciences he switched to mathematics, but he kept a great interest in the applications of mathematics in physics and mechanics. He wrote a thesis "On the propagation of waves in elastic tubes" under van der Waals and defended it on July 12, 1878. The University of Amsterdam had just been granted the right to confer doctorates, and so Korteweg became the first doctor of our university. In fact the university existed already since 1632, but only as Atheneum Illustre . Three years later Korteweg was appointed at the University of Amsterdam as professor of mathematics, mechanics and astronomy. In his inaugural address he stressed the importance of mathematical applications in the sciences.

His influence on academic life in the Netherlands becomes apparent also from his membership of several academic institutions; he was a member of the Royal Academy for sixty years and of the Wiskundig Genootschap (Dutch Mathematical Society) for seventy five years. As an editor of the `Nieuw Archief voor Wiskunde' during the period from 1897 to 1941 he contributed greatly to the development of mathematics in the Netherlands.

After a fruitful life D.J. Korteweg passed away at age ninety three on the tenth of May 1941.

Gustav de Vries

Life of G. de Vries

Gustav de Vries was born January 22, 1866 in Amsterdam. He studied in Amsterdam under van der Waals, Julius, Pesch and Korteweg. The latter also became his thesis advisor. He worked on his thesis while being employed as a teacher at the KMA (Royal Military Academy) in Breda (1892-1893) and at the "cadettenschool" te Alkmaar, (1893-1894). De Vries defended his thesis

Bijdrage tot de kennis der lange golven, Acad. proefschrift, Universiteit van Amsterdam, 1894, 95 pp, Loosjes, Haarlem.

at the University of Amsterdam, December 1, 1894. Shortly afterwards the main results of the thesis were made public in the famous Korteweg - de Vries paper (see paper in pdf, courtesy of Bijzondere Collecties, Universiteit van Amsterdam, UBM: DT 9721):

On the Change of Form of Long Waves advancing in a Rectangular Canal and on a New Type of Long Stationary Waves; Philosophical Magazine, 5th series, 39, 1895, pp. 422–443

He served as a high school teacher at the ``HBS en Handelsschool'' of Haarlemfrom 1894 to 1931.

De Vries was married to Johanna Boelen, who taught French language and literature. They had five children. Gustav de Vries died in Haarlem, 16 December 1934.

Scientific work of D.J. Korteweg

As a student at the University of Amsterdam Korteweg was impressed by the work of the later Nobel-laureate J.D. van der Waals (the equation of state and the continuity of the gas and fluid phases), and he published a paper on thermodynamics related to his work. Van der Waals also became his thesis supervisor. Korteweg was appointed at the University of Amsterdam as professor of mathematics, mechanics and astronomy in 1881. He became the first full professor of mathematics. He had stressed the importance of mathematical applications in the sciences in his inaugural address. His main interest was indeed in that direction and he worked together with, among others, van der Waals and van 't Hoff; he wrote papers in the fields of classical mechanics, fluid mechanics and thermodynamics. These researches led him also to pure mathematics; we mention his investigations on algebraic equations with real coefficients and his study on the properties of surfaces in the neighbourhood of singular points.

Although much of this work lies now in the shadows of history, there is one subject that still attracts the attention of hundreds of mathematicians, physicists, chemists and engineers, namely the theory of long stationary waves and the famous Korteweg-de Vries equation. This equation has become the source of important breakthroughs in mechanics and nonlinear analysis and of many developments in algebra, geometry and physics.

In one of his treatises on hydrodynamics Sir Horace Lamb stated that even when friction is neglected, long waves in a canal with rectangular cross section must necessarily change their form as they advance, becoming steeper in front and less steep behind. Because of earlier investigations of Boussinesq, Lord Raleigh and Saint-Venant, the truth of this assertion was not generally accepted, but it seemed to Korteweg that many authors were inclined to believe that a so-called stationary wave without change of form was only stationary to a certain approximation. Whatever the opinion of the mathematical community in those days, Korteweg and his student G. de Vries settled the question of the existence of stationary waves in the latter's doctoral thesis , and a year later in their famous paper `On the Change of Form of Long Waves advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves'. Here the conclusion was drawn that in a frictionless liquid there may indeed exist absolutely stationary waves. In a special case these waves take the form of one or more separated heaps of water propagating with a velocity proportional to their amplitude. The larger ones may overtake the smaller ones and when this happens the waves interchange position without changing their form. They may be compared with colliding marbles exchanging their momentum, reason why Kruskal and Zabusky later called them `solitons'.

Another scientific achievement of Korteweg is his edition of the `Oeuvres Complètes' of the mathematical physicist `avant la lettre' Christiaan Huygens under the auspices of the `Hollandsche Maatschappij der Wetenschappen'; he was the principal leader of the project during the period 1911-1927. Any scientist interested in the history of his field knows the mental exertion required to understand the way of thinking and reasoning of his predecessors.

Students and teaching of D.J. Korteweg

Korteweg inspired many young mathematicians who wrote their doctoral thesis under his supervision. Besides G. de Vries we mention Hk. de Vries, H.J.E. Beth (the father of the logician E.W. Beth) and last but not least, L.E.J. Brouwer.

The conscientiousness and the courage shown by Korteweg, albeit an applied mathematician, in supervising Brouwer's thesis `Over de Grondslagen van de Wiskunde' (`On the foundations of Mathematics') is remarkable. The thesis was defended in 1907 and the negation of the `principle of the excluded middle' followed in 1908. Ten years later the thunder was roaring between Göttingen and Amsterdam.

Korteweg's teaching duties concerned analytic and projective geometry, mechanics, astronomy and probability theory. He was meticulous in the preparation of his lectures and keenly interested in the progress of his students; a student could have a tough time when he was not applying himself.

Eduard de Jager

References

  • H.J.E. Beth, W. van der Woude, Levensbericht van D. J. Korteweg; Jaarboek van de Kon. Akad. v. Kunsten en Wetensch., 1945-1946, 194-208
  • B. Willink, De tweede Gouden Eeuw; B. Bakker, Amsterdam, 1998
  • D.J. Korteweg
    • Over de voortplantingsnelheid van golven in elastische buizen, Acad. proefschrift, Universiteit van Amsterdam, 1878, 166 pp. van Doesburgh, Leiden.
    • De wiskunde als hulpwetenschap; Intreerede, Univ. van Amsterdam 1881
    • Einfluss der räumlichen Ausdehnung der Moleküle auf den Druck eines Gases; Annalen der Physik, 1881
    • Algemene stellingen betreffende de stationaire beweging eener onsamendrukbare wrijvende vloeistof; Verslagen en mededelingen der Kon. Ak. v. Wetensch. Afd. Natuurk., 2e reeks, 18, 1883, pp. 343–359.
    • Over de banen beschreven onder den invloed eener centrale kracht; Mededelingen Kon. Ak. v. Wetensch., 2e reeks, 20 1884, pp. 247–289.
    • Über stabilität periodischer ebener Bahnen; Sitzungsberichte Akad. Wien, 1886.
    • D.J. Korteweg, Über Faltenpunkte; Sitzungsberichte der Akademie der Wissenschaften Wien, Mathematisch-Naturwissenschafliche Klasse, Abteilung 2A (1889), pp.1154-1191.
    • D.J. Korteweg, Sur les points de plissement [On plait points], Archives Néerlandaises des Sciences Exactes et Naturelles; Société Hollandaise des Sciences, volume 24, (1891) pp. 57–98.
    • La théorie générale des plis et la surface ψ de van der Waals dans le cas de symétrie; Archives Néerlandaises des Sci. Exactes et Naturelles; Société Hollandaise des Sciences, 24, 1891, pp. 295–368.
    • Über Singularitäten verschiedener Ausnahme-Ordnung und ihre Zerlegung; Mathematische Annalen; 41, 1893, pp. 286–307.
    • with G. de Vries; On the Change of Form of Long Waves advancing in a Rectangular Canal and on a New Type of Long Stationary Waves; Philosophical Magazine, 5th series, 39, 1895, pp. 422–443.
      See paper in pdf (courtesy of Bijzondere Collecties, Universiteit van Amsterdam, UBM: DT 9721).
      Read a review JFM 26.0881.02 (courtesy of Zentralblatt MATH).
    • Sur certaines vibrations d'ordre supérieure et d'intensité anormale; Archives Néerlandaises des Sci. Exactes et de Nature, 2e sér., 1, 1897, pp. 229-260.
    • Über eine ziemlich verbreitete unrichtige Behandlungsweise der rollenden Bewegung und ins Besondere über kleine rollende Schwingungen um eine Gleichgewichtslage; Nieuw Archief voor Wiskunde, 2e reeks, 4, 1899-1900, pp. 130–162.
    • Sur un théorème remarquable, qui se rapporte à la théorie des équations algébriques à paramètres réels, dont toutes les racines restent constamment réelles; Nieuw Archief voor Wiskunde, 2e reeks, 4, 1899–1900, pp. 46–54.
    • Over de verschillende evenwichtsstanden van drijvende rechthoekig-parallellepidische lichamen, wier lengteas met de vloeistofoppervlakte evenwijdig loopt; Nieuw Archief voor Wiskunde, 2e reeks, 8, 1907–1909, pp. 1–25.
    • Huygens' sympathetic clocks and related phenomena; Proceedings Royal Acad., Amsterdam, 8, 1905, pp. 436–455.
  • E. van Groesen, E.M. de Jager; Mathematical Structures in Continuous Dynamical Systems; Studies in Mathematical Physics, 6, 617pp., North Holland Publishing Cy, Amsterdam, 1994.
  • A.S. Fokas, V.E. Zakharov (eds), Important Developments in Soliton Theory; Springer Series in Nonlinear Dynamics, Springer, Berlin, pp. 559, 1993.
  • M. Hazewinkel, H.W. Capel, E.M. de Jager (eds); Proceedings of the International Symposium KdVI'95, to commemorate the centennial of the equation by and named after Korteweg and de Vries, Kluwer Academic Publishers, pp 516, 1995.
  • H. Lamb; Hydrodynamics, 1895; 6th edition Cambridge University Press, 1932.
  • J. de Boussinesq; Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal en communiquant au liquide continu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures et Appliquées 2, 1872, p 55.
  • N.J. Zabusky, M.D. Kruskal, Interaction of Solitons in a Collisionless Plasma and the Recurrence of Initial States; Physical Review Letters, 15, no 6, 1965.
  • F. van der Blij; Some details of the history of the Korteweg-de Vries equation; Nieuw Archief voor Wiskunde (3) 26, 1978, pp. 54–64.
    See paper in pdf.
  • G. de Vries, Bijdrage tot de kennis der lange golven, Acad. proefschrift, Universiteit van Amsterdam, 1894, 95 pp, Loosjes, Haarlem.
  • Hk. de Vries; Over de restdoorsnede van twee volgens eene vlakke kromme perspectivische kegels, en over satelliet krommen, Acad. proefschrift Universiteit van Amsterdam, 1901, 150 pp, Delsman en Nolthenius, Amsterdam.
  • H.J.E. Beth; De schommelingen om een evenwichtsstand bij het bestaan eener eenvoudige lineaire relatie tussen de reciproke waarden der perioden, met toepassing op de beweging, zonder wrijving, van een zwaartepunt op den bodem eener vaas; Acad. proefschrift Universiteit van Amsterdam, 1910, 135 pp., Kok, Kampen.
  • L.E.J. Brouwer; Over de Grondslagen van de Wiskunde, Acad. proefschrift Universiteit van Amsterdam, 1907, 183 pp., Maas en van Suchtelen, Amsterdam, Leipzig.
  • D. van Dalen, L.E.J. Brouwer; Over de Grondslagen van de Wiskunde; M.C. Varia; Mathematical Centre C.W.I., Amsterdam 1981, 267 pp.
  • D.van Dalen, Mystic, Geometer, Intuitionist; the life of L.E.J. Brouwer, vol. I, the Dawning Revolution, 440 pp., Clarendon Press, Oxford, 1999.
  • Christiaan Huygens; Oevres Complètes I.XV; Hollandsche Maatschappij der Wetenschappen.
  • Levelt Sengers, Johanna, and Levelt, Antonius H.M., Diederik Korteweg, pioneer of criticality, Physics Today Vol. 55, pp. 47-53, Dec 2002.