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).

[2] 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.

[3][4] 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.

[5] The KdV equation is a partial differential equation that models (spatially) one-dimensional nonlinear dispersive nondissipative waves described by a function

in front of the last term is conventional but of no great significance: multiplying

, eventually slides down to the local minimum, then back up the other side, reaching an equal height, and then reverses direction, ending up at the local maximum again at time

This is the characteristic shape of the solitary wave solution.

[8] The solution depends on a set of decreasing positive parameters

The KdV equation has infinitely many integrals of motion, functionals on a solution

are defined recursively by The first few integrals of motion are: Only the odd-numbered terms

result in non-trivial (meaning non-zero) integrals of motion.

[11] The Lax pair accounts for the infinite number of first integrals of the KdV equation.

is the time-independent Schrödinger operator (disregarding constants) with potential

It can be shown that due to this Lax formulation that in fact the eigenvalues do not depend on

defined by Since the Lagrangian (eq (1)) contains second derivatives, the Euler–Lagrange equation of motion for this field is where

, so use that to simplify the above terms, Finally, plug these three non-zero terms back into eq (3) to see which is exactly the KdV equation It can be shown that any sufficiently fast decaying smooth solution will eventually split into a finite superposition of solitons travelling to the right plus a decaying dispersive part travelling to the left.

This was first observed by Zabusky & Kruskal (1965) and can be rigorously proven using the nonlinear steepest descent analysis for oscillatory Riemann–Hilbert problems.

[14] The history of the KdV equation started with experiments by John Scott Russell in 1834, followed by theoretical investigations by Lord Rayleigh and Joseph Boussinesq around 1870 and, finally, Korteweg and De Vries in 1895.

The KdV equation was not studied much after this until Zabusky & Kruskal (1965) discovered numerically that its solutions seemed to decompose at large times into a collection of "solitons": well separated solitary waves.

Moreover, the solitons seems to be almost unaffected in shape by passing through each other (though this could cause a change in their position).

They also made the connection to earlier numerical experiments by Fermi, Pasta, Ulam, and Tsingou by showing that the KdV equation was the continuum limit of the FPUT system.

Development of the analytic solution by means of the inverse scattering transform was done in 1967 by Gardner, Greene, Kruskal and Miura.

[2][15] The KdV equation is now seen to be closely connected to Huygens' principle.

[16][17] The KdV equation has several connections to physical problems.

In addition to being the governing equation of the string in the Fermi–Pasta–Ulam–Tsingou problem in the continuum limit, it approximately describes the evolution of long, one-dimensional waves in many physical settings, including: The KdV equation can also be solved using the inverse scattering transform such as those applied to the non-linear Schrödinger equation.

real, one obtains an attractive self-interaction that should yield a bright soliton.

[citation needed] Many different variations of the KdV equations have been studied.