Korteweg-de Vries Institute for Mathematics

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 Compl├Ętes' 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.

Published by  Korteweg de Vries Institute

4 September 2014