REVIEW
On the origin of the Korteweg-de Vries equation
History and Overview · Submitted 28 Feb 2006 · 31 references
Pith's verdict: A mathematically sound historical essay whose central attribution claim—Boussinesq wrote KdV avant la lettre—needs primary-source verification, but the derivation and the comparison of boundary conditions are genuinely useful.
T0 · AI review
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
The Korteweg-de Vries equation was already in Boussinesq's 1872 equations, waiting on a substitution he never performed.
The reading
What carries the argument
At the center is a substitution identity. Boussinesq's continuity equation (4.5), $\frac{\partial h}{\partial t} + \frac{\partial}{\partial x}(\omega h) = 0$, together with his pointwise wave velocity (4.8), $\omega = \sqrt{gH} + \sqrt{gH}\left(\frac{3h}{4H} + \frac{H^2}{6h}\frac{\partial^2 h}{\partial x^2}\right)$, yields (4.11), $\frac{\partial h}{\partial t} + \sqrt{\frac{g}{H}}\,\frac{3}{2}\,\frac{\partial}{\partial x}\left(\frac{2}{3}Hh + \frac{1}{2}h^2 + \frac{H^3}{9}\frac{\partial^2 h}{\partial x^2}\right) = 0$, which is the KdV equation. The paper also invokes the auxiliary function $\psi_1$ from the 1877 footnote, and the factorization (4.12) of the bidirectional Boussinesq equation into two KdV-type factors. These algebraic moves are what convert a historical comparison into an equivalence of equations.
Extended reading notes
Core claim
On the paper's own terms, Boussinesq's analysis contains the KdV equation twice, in two different senses. First, the displayed equations (4.5) and (4.8) of his 1872 article combine by direct substitution into equation (4.11), which is the Korteweg-de Vries equation in a fixed coordinate frame; Boussinesq did not make that substitution. Second, according to the paper, a footnote on page 360 of the 1877 'Essai sur la théorie des eaux courantes' effectively writes out the KdV equation via an auxiliary function $\psi_1$, a fact recovered only in the modern historical literature. The paper adds that the bidirectional Boussinesq equation factorizes into two unidirectional factors, one of which is KdV, as shown in equation (4.12). What Korteweg and De Vries added was not the equation itself but a derivation that avoids the vanishing-at-infinity assumption, so the same equation governs periodic cnoidal waves as well as solitary waves.
Load-bearing premise
The historical-priority claim depends on the report that Boussinesq's 1877 footnote actually writes out the Korteweg-de Vries equation; if that report is wrong, the algebraic derivation from his other equations still stands, but the claim of an explicit earlier statement does not.
Editorial extensions
If this is right
- If the substitution is legitimate, Boussinesq's 1872 article contains the KdV equation in unidirectional form, and his 1877 memoir contains it explicitly in a footnote; the customary name is then a matter of historical reception rather than first derivation.
- The 1895 Korteweg-de Vries equation is equivalent to Boussinesq's solitary-wave theory when surface tension is neglected; the remaining difference is the choice of a fixed versus a moving coordinate frame.
- Because the bidirectional Boussinesq equation factorizes into two unidirectional factors, the KdV equation can be read as one half of a more general two-directional shallow-water model.
- Korteweg and De Vries's distinctive contribution is the removal of the vanishing-at-infinity assumption, which lets one equation govern both solitary waves and periodic cnoidal waves; Boussinesq treated periodic waves separately.
- Boussinesq's conserved 'moment de stabilité' appears inside a Hamiltonian formulation of the KdV equation, so his stability argument anticipates the later Hamiltonian theory of integrable wave equations.
Reading between the lines
- A reader can infer that the equation's name is partly a historical accident: had the 1877 footnote been incorporated into the standard narrative earlier, the central object of soliton theory could plausibly be called the Boussinesq equation.
- The paper's method—checking whether a displayed equation in an older memoir turns into a later equation by a simple substitution—offers a general historiographic test, provided one also checks the validity conditions under which the two equations live.
- A natural follow-up question, beyond the historical claim, is how much of the KdV equation's integrable structure (soliton collisions, infinite conservation laws) is visible already in the bidirectional Boussinesq equation from which KdV factors out.
Original abstract
The Korteweg-de Vries equation has a central place in a model for waves on shallow water and it is an example of the propagation of weakly dispersive and weakly nonlinear waves. Its history spans a period of about sixty years, starting with experiments of Scott Russell in 1834, followed by theoretical investigations of, among others, Lord Rayleigh and Boussinesq in 1871 and, finally, Korteweg and De Vries in 1895. In this essay we compare the work of Boussinesq and Korteweg-de Vries, stressing essential differences and some interesting connections. Although there exist a number of articles, reviewing the origin and birth of the Korteweg-de Vries equations, connections and differences, not generally known, are reported.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The Korteweg-de Vries equation has a central place in a model for waves on shallow water and it is an example of the propagation of weakly dispersive and weakly nonlinear waves. Its history spans a period of about sixty years, starting with experiments of Scott Russell in 1834, followed by theoretical investigations of, among others, Lord Rayleigh and Boussinesq in 1871 and, finally, Korteweg and De Vries in 1895. In this essay we compare the work of Boussinesq and Korteweg-de Vries, stressing essential differences and some interesting connections. Although there exist a number of articles, reviewing the origin and birth of the Korteweg-de Vries equations, connections and differences, not generally known, are reported.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Korteweg-de Vries equation has a central place in a model for waves on shallow water and it is an example of the propagation of weakly dispersive and weakly nonlinear waves. | background 1.00 | 0.05 | 0.04 | asserted 1.00 |
| s1 | Its history spans a period of about sixty years, starting with experiments of Scott Russell in 1834, followed by theoretical investigations of, among others, Lord Rayleigh and Boussinesq in 1871 and, finally, Korteweg and De Vries in 1895. | background 1.00 | 0.03 | 0.03 | asserted 1.00 |
| s2 | In this essay we compare the work of Boussinesq and Korteweg-de Vries, stressing essential differences and some interesting connections. | contribution 1.00 | 0.72 | 0.09 | asserted 0.91 |
| s3 | Although there exist a number of articles, reviewing the origin and birth of the Korteweg-de Vries equations, connections and differences, not generally known, are reported. | gap 0.66 | 0.57 | 0.12 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:math/0602661 by the authors.
References
How the literature uses it
4 arXiv papers cite this in 4 sentences
Papers citing it
Cites it as background 4 papers
-
The Korteweg-de Vries (KdV) equation [ref] is an essential tool in physics thanks to its wide range of applications [ref] — most notably in fluid dynamics [ref] — and its powerful integrability properties [ref] .
Orbital Bifurcations and Shoaling of Cnoidal Waves · Introduction · 1.00
-
See [ref] for a discussion on the origin of the KdV equation.] equation [ref] :
Integrability in Perturbed Black Holes: Background Hidden Structures · Darboux transformation and integrability of the KdV equation · 0.97
-
The equation ([ref]) is called the Korteweg-de Vries equation in the literature, often abbreviated as the KdV equation, although it had appeared explicitly in Boussinesq’s massive 1877 Memoir [ref] , as equation (283bis) in a footnote on page 360[The interested readers are referred to [ref] for history and origins of the Korteweg-de Vries equation.].
Initial boundary value problem for Korteweg-de Vries equation: a review and open problems · Introduction · 0.95
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.