Korteweg-de Vries Equation

The partial differential equation

(1)

(Lamb 1980; Zwillinger 1997, p. 175), often abbreviated "KdV." This is a nondimensionalized version of the equation

(2)

derived by Korteweg and de Vries (1895) which described weakly nonlinear shallow water waves. Here, , is the channel height, is the surface tension, is the gravitational acceleration, and is the density. This equation was found to have solitary wave solutions, vindicating the observations made 51 years earlier of a solitary channel wave by Russell in Aug. 1834 (Russell 1844).

It is a little-known fact that the first genus-2 solution to the Korteweg-de Vries equation was given by Baker (1907; Previato 2004).

Zabusky and Kruskal (1965) subsequently studied the continuum limit of the Fermi-Pasta-Ulam Experiment and, surprisingly, obtained the Korteweg-de Vries equation. They found that the solitary wave solutions had behavior similar to the superposition principle, despite the fact that the waves themselves were highly nonlinear. They dubbed such waves solitons, and proceeded to devise new solution technique for them (Miura et al. 1968). Miura et al. (1968) found nine conservation laws and Miura (1968) found a tenth, hinting that an infinite number of conserved quantities might exist (Tabor 1989, p. 288). In fact, a transformation due to Gardner provides an algorithm for computing an infinite number of conserved densities of the KdV equation, which are connected to those of the so-called modified KdV equation through the Miura transformation

(3)

(Tabor 1989, p. 291). The Korteweg-de Vries equation also exhibits Galilean invariance.

An important step in the solution of the KdV equation was provided by Gardner et al. (1967), who proposed that it could be studied through the properties of the one-dimensional Schrödinger equation for potential

(4)

obtained by making a variable substitution in (3) and using Galilean invariance. If the corresponding quantum mechanical inverse scattering problem (i.e., going from the associated quantum mechanical properties--termed scattering data--to the potential) can be solved, the evolution of could then be reconstructed without having to actually solve the KdV equation (Tabor 1989, pp. 291-292). While this procedure sounds complicated, and in fact can only be solved exactly for rather special cases, it can be viewed as a more complicated analog of inverse Fourier transforms (which turns out is known as an inverse scattering transform). Using inverse scattering transforms, -soliton solutions can be obtained.

Lax (1968) showed that the KdV equation is equivalent to the so-called "isospectral integrability condition" for pairs of linear operators, known as Lax pairs (Tabor 1989, p. 304).

The so-called generalized KdV equation is given by

(5)

(Boyd 1986; Zwillinger 1997, p. 175). The so-called deformed KdV equation is given by

(6)

(Dodd and Fordy 1983; Zwillinger 1997, p. 178), and the modified KdV equation is given by

(7)

(Calogero and Degasperis 1982, p. 51; Tabor 1989, p. 304; Zwillinger 1997, p. 178), or

(8)

(Dodd and Fordy 1983; Zwillinger 1997, p. 178).

The cylindrical KdV equation is given by

(9)

(Calogero and Degasperis 1982, p. 50; Zwillinger 1997, p. 175), and the spherical KdV by

(10)

(Calogero and Degasperis 1982, p. 51; Zwillinger 1997, p. 175).