Korteweg–de Vries equation

The Korteweg -de Vries equation ( KdV ) is a nonlinear partial differential equation of the third order. It was proposed in 1895 by Diederik Korteweg and Gustav de Vries for the analysis of shallow water waves in narrow channels, but it has been already examined by Boussinesq in 1877. It describes solitons were first observed in water channels in 1834 by John Scott Russell. 1965 could Norman Zabusky and Martin Kruskal, the quasi - periodic behavior in the Fermi - Pasta - Ulam experiment explain by showing that the KdV equation is the continuous limit.

Mathematical formulation

The CO equation is formulated as a partial differential equation in one dimension x and time t. She is a third order equation. Originally, it was by Korteweg and de Vries in the form

With specifically formulated for waves in channels, where L indicates the depth, g is the gravitational acceleration, T is the surface tension and ρ is the density of the liquid. In today's literature, however, one finds the equation mostly in the abstract form

Which is derivable by several transformation steps from the original equation. One of the important properties is the existence of Solitonenlösungen. The simplest of these

Wherein any constants that describes a single soliton with right running speed.

Mathematical Methods

The KdV equation is an example of a completely integrable system. So you can solve the KdV equation developed by Clifford Gardner, John Greene, Martin Kruskal and Miura Robert inverse scattering transform: For this one arranges a one-dimensional Schrödinger operator of a solution

About. This, together with the operator

The Lax pair of the KdV equation. That is, if and only solves the KdV equation when

Applies. Also, you can assign the Schrödinger operator, the scattering data (reflection coefficient and eigenvalues ​​plus normalization constants). The eigenvalues ​​are independent of time due to the Lax equation. Reflection coefficient and normalization constants satisfy linear differential equations which can be solved explicitly. The solution is then reconstructed by inverse scattering theory.

This has some interesting consequences. On one hand, we obtain that solutions of the KdV equation exist for all time, on the other hand, one obtains that the solitons correspond exactly to the eigenvalues. Moreover, one can show that any, sufficiently strongly sloping initial conditions are asymptotically for large times t is given by a finite number to the right of current solos tions and a running left dispersive component.