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Wikipedia / 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 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.

In 1970, a generalization of the one-dimensional Korteweg–de Vries (KdV) equation to two spatial dimensions, x and y, was introduced by Kadomtsev and Petviashvili (see Kadomtsev–Petviashvili equation). The stability and dynamics of the KdV type solitons with respect to nonlinear transverse perturbations of finite wavelength are described by the Shrira-Pesenson equation.

Definition

The KdV equation is a partial differential equation that models (spatially) one-dimensional nonlinear dispersive nondissipative adhering to:

is an advection term.

For modelling shallow water waves is the height displacement of the water surface from its equilibrium height.

by constants can be used to make the coefficients of any of the three terms equal to any given non-zero constants.

Soliton solutions

One-soliton solution

Consider solutions in which a fixed waveform, maintains its shape as it travels to the right at phase speed. Substituting it into the KdV equation gives the ordinary differential equation

,

is a constant of integration satisfies Newton's equation of motion of a particle of unit mass in a cubic potential

  1. .

If

has local maximum, 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. This is the characteristic shape of the solitary wave solution.

More precisely, the solution is

stands for the hyperbolic secant is an arbitrary constant. This describes a right-moving soliton.

N-soliton solution

separate single solitons.

This is derived using the inverse scattering method.

Integrals of motion

The KdV equation has infinitely many integrals of motion which do not change with time. They can be given explicitly as

are defined recursively by

The first few integrals of motion are:

  • .

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

Lax pairs

The KdV equation

can be reformulated as the Lax equation

a Sturm–Liouville operator:

is the commutator. The Lax pair accounts for the infinite number of first integrals of the KdV equation.

is the time-independent Schrödinger operator.

Zero-curvature representation

Setting the components of the Lax connection

Least action principle

The Korteweg–De Vries equation

is the Euler–Lagrange equation of motion derived from the Lagrangian density

defined by

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

component.

is implied so eq (2) really reads,

Evaluate the five terms of eq (3) by plugging in eq (1),

, 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

Long-time asymptotics

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 and can be rigorously proven using the nonlinear steepest descent analysis for oscillatory Riemann–Hilbert problems.

History

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 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.

The KdV equation is now seen to be closely connected to Huygens' principle.

Applications and connections

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 similarly to the non-linear Schrödinger equation.

KdV equation and the Gross–Pitaevskii equation

Considering the simplified solutions of the form

we obtain the KdV equation as

or

Integrating and taking the special case in which the integration constant is zero, we have:

special case of the generalized stationary Gross–Pitaevskii equation (GPE)

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

Variations

Many different variations of the KdV equations have been studied. Some are listed in the following table.

See also

Notes

References