D.J. Korteweg and G. de Vries
Notes by Eduard de Jager, slightly edited by Tom Koornwinder.
A version of these notes is also available on the
site of the Kortewegde Vries Institute.
Life of D. J. Korteweg
In the memorial on Korteweg 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 and
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 socalled
"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.
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 (18921893) and at the "cadettenschool" te Alkmaar,
(18931894). 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 Haarlem from 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
Kortewegde 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 SaintVenant, the truth of this assertion was not
generally accepted, but it seemed to Korteweg that many authors were
inclined to believe that a socalled 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 in Philosophical Magazine.
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 19111927. 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.
References
 H.J.E. Beth, W. van der
Woude, Levensbericht van D. J. Korteweg; Jaarboek van de
Kon. Akad. v. Kunsten en Wetensch., 19451946, 194208
 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, MathematischNaturwissenschafliche
Klasse, Abteilung 2A (1889), pp.11541191.
 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 AusnahmeOrdnung 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. 229260.
 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, 18991900, 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
rechthoekigparallellepidische 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 Co, 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 Kortewegde 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. 4753,
Dec 2002.
Links
 E.M. de Jager,
On the origin of the Kortewegde Vries equation,
Forum der Berliner Mathematischen Gesellschaft, Band 19, Dezember 2011, pp.
171195;
arXiv:math/0602661v2 [math.HO].

J. Levelt Sengers,
How Fluids Unmix:
Discoveries by the School of Van Der Waals and Kamerlingh Onnes,
KNAW Edita, 2002;
University of Chicago Press.
There is also a free digital version at the page
History of Science and Scholarship in the Netherlands
(2003 Volume 4).

An Introduction to Solitons (by Alex Kasman)

The History and Significance of the KdVI Equation (by Alex Kasman)

Solitons Home Page at HeriotWatt

Many Faces of Solitons (by Kanehisa Takasaki)

Van golven in ondiep water tot intersectietheorie op moduliruimten
(in Dutch, slides bij de oratie van Sergey Shadrin, UvA,
6 december 2013)

Line soliton interactions observed on flat beaches

Short biographies of mathematicians
(MacTutor History of Mathematics archive):

Nobel Prizes