Korteweg–De Vries equation

In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE, exhibiting typical behaviors such as a large number of explicit solutions, in particular soliton solutions, and an infinite number of conserved quantities, despite the nonlinearity which typically renders PDEs intractable. The KdV can be solved by the inverse scattering method (ISM). In fact, Clifford Gardner, John M. Greene, Martin Kruskal and Robert Miura developed the classical inverse scattering method to solve the KdV equation.

The KdV equation was first introduced by Joseph Valentin Boussinesq (1877, footnote on page 360) and rediscovered by Diederik Korteweg and Gustav de Vries in 1895, who found the simplest solution, the one-soliton solution. Understanding of the equation and behavior of solutions was greatly advanced by the computer simulations of Norman Zabusky and Kruskal in 1965 and then the development of the inverse scattering transform in 1967.

In 1972, T. Kawahara proposed a fifth-order KdV type of equation, known as Kawahara equation, that describes dispersive waves, particularly in cases when the coefficient of the KdV equation becomes very small or zero.