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Korteweg-de Vries equation (KdV)

Korteweg-de Vries equation
Order	3rd
Solvable	Exactly solvable
The Cauchy problem	Solvable by inverse scattering transform
Hierarchy	Korteweg-de Vries hierarchy
First introduced	Korteweg & de Vries, 1895

1 Korteweg-de Vries equation (KdV)

Definition

$u_{t} + 6 u u_{x} + u_{x x x} = 0$

History

The equation is named for Diederik Korteweg and Gustav de Vries who studied it in 1895 (Korteweg & de Vries, 1895), though the equation first appears in work of Boussinesq (1877). The Korteweg-de Vries equation was not studied much after that until Zabusky and Kruskal (1965) discovered numerically that its solutions seemed to decompose at large times into a collection of solitons.

Solutions

Traveling wave

The Korteweg-de Vries equation possesses a solution known from the end of the 19^th century. This solution can be expressed via Jacobi's elliptic functions. Let us look for solutions of the Korteweg-de Vries equation

$u_{t} + 6 u u_{x} + u_{x x x} = 0$

in the form of traveling wave

$u (x, t) = y (z), z = x - c_{0} t .$

Then the Korteweg-de Vries equation will take the form

$y_{z z} + 3 y^{2} - c_{0} y + c_{1} = 0.$

Details

Multiplying the latter equation by

y_{z}

and integrating the result with respect to

z

we get

$\frac{y_{z}^{2}}{2} + y^{3} - \frac{c_{0} y^{2}}{2} + c_{1} y + c_{2} = 0.$

This equation can be written in the form

$y_{z}^{2} = - 2 f (y), f (y) = y^{3} - \frac{c_{0} y^{2}}{2} + c_{1} y + c_{2} .$

Function

f (y)

can be written in the form

$f (y) = (y - α) (y - β) (y - γ)$

where

α

β

and

γ

are real roots of cubic equation

$f (y) = y^{3} - \frac{c_{0} y^{2}}{2} + c_{1} y + c_{2} = 0.$

One can express

α

β

and

γ

via

c_{0}

c_{1}

and

c_{2}

$α β γ = - c_{2},$

$α β + α γ + β γ = c_{1},$

$α + β + γ = \frac{c_{0}}{2} .$

So we obtain

$\frac{d y}{\sqrt{2 (α - y) (y - β) (y - γ)}} = d z .$

Let us introduce new variables

$y = α - p^{2}, p = \sqrt{α - β} q, S^{2} = \frac{α - β}{α - γ} .$

Using these variables we get

$\int_{0}^{q} \frac{d x}{\sqrt{(1 - x^{2}) (1 - S^{2} x^{2})}} = \sqrt{\frac{α - γ}{2}} (z - z_{0}) .$

In the left hand side of the latter expression one can notice the elliptic integral of the first kind

$F (\arcsin q, S) = s n^{- 1} (q, S) = \sqrt{\frac{α - γ}{2}} (z - z_{0}) .$

Taking this into account we have the following solution

$q = s n {\sqrt{\frac{α - γ}{2}} (z - z_{0}), S^{2}},$

where

sn (z)

is the Jacobi elliptic function. Returning from

q

y

we obtain

$y (z) = α - (α - β) s n^{2} {\sqrt{\frac{α - γ}{2}} (z - z_{0}), S^{2}} .$

Now let us take into account the well-known relation between elliptic functions

$s n^{2} (z) + c n^{2} (z) = 1$

One can write solution of the Korteweg-de Vries equation in the form

$u (x, t) = β + (α - β) c n^{2} {\sqrt{\frac{α - γ}{2}} (z - z_{0}), S^{2}}$

with

0 \leq S \leq 1

This solution is a wave with speed

c_{0}

and period

T

defined by the following expressions

$c_{0} = 2 (α + β + γ),$

$T = 2 \sqrt{\frac{2}{α - γ}} \int_{0}^{1} \frac{d x}{\sqrt{(1 - x^{2}) (1 - S^{2} x^{2})}} = \sqrt{\frac{8}{α - γ}} K (S),$

where

K (S)

is the complete elliptic integral of the first kind.

Thus we obtained periodic wave described by the Korteweg-de Vries equation. This wave sometimes is referred to as cnoidal wave for the look of the corresponding Jacobi elliptic function sign. In limiting case of small amplitude this solution turns into the well-known sinusoidal wave.

Soliton

If in the obtained periodic solution

α > β = γ

then

S^{2} = 1

and the period

T

tends to infinity and we obtain solitary wave

$u (x, t) = y (z) = β + (α - β) c h^{- 2} {\sqrt{\frac{α - β}{2}} (z - z_{0})} .$

Usually one sets

β = γ = 0

α = 2 k^{2}

so the solution takes the form

$u (x, t) = 2 k^{2} c h^{- 2} {k (x - 4 k^{2} t) + χ_{0}},$

where

χ_{0}

is an arbitrary constant.

This solution describes the solitary wave observed by John Scott Russel in 1834 (Russel, 1845).

N-soliton solutions

N-soliton solutions of the Korteweg-de Vries equation can be obtained with the help of the Hirota direct method.

See Application of Hirota method for the Korteweg-de Vries equation.

Self-similar solutions

Let us look for self-similar solutions of the Korteweg-de Vries equation

$u_{t} + 6 u u_{x} + u_{x x x} = 0$

in the form

$u (x, t) = \frac{1}{{(3 t)}^{\frac{2}{3}}} y (z), z = \frac{x}{{(3 t)}^{\frac{1}{3}}} .$

Details

We then obtain reduction of the Korteweg-de Vries equation to the following ordinary differential equation

$y_{z z z} + 6 y y_{z} - z y_{z} - 2 y = 0.$

Now let us use the Miura transform

$y (z) = w_{z} - w^{2}, w = w (z) .$

For new unknown function one obtains

$E_{1} [w] = w_{z z z z} - 2 w w_{z z z} - 12 w w_{z}^{2} - 6 w^{2} w_{z z} + 12 w^{3} w_{z} -$

$- z w_{z z} + 2 z w w_{z} - 2 w_{z} + 2 w^{2} = 0.$

The latter equation can be written in the form

$E_{1} [w] = (\frac{d}{d z} - 2 w) \frac{d}{d z} (\frac{d^{2} w}{d z^{2}} - 2 w^{3} - z w + α) = 0.$

We obtain that solution of the Korteweg-de Vries equation can be expressed via solutions of equation

$w_{z z} - 2 w^{3} - z w + α = 0.$

This is the second Painlevé equation. Solutions of this equation generally can be expressed via the so-called Painlevé transcendents. For integer

α

the second Painlevé equation possesses rational solutions; for half-integer

α

its solutions can be expressed via Airy functions.

Thus we've found out that self-similar solutions of the Korteweg-de Vries equation can be expressed via solutions of the second Painlevé equation.

Integrals of motion

Dicovery of solitons had put a question about integrals of motion for the Korteweg-de Vries equation

$u_{t} + 6 u u_{x} + u_{x x x} = 0 .$

Details

This equation can be presented in the form

$\frac{\partial u}{\partial t} + \frac{\partial}{\partial x} (u_{x x} + 3 u^{2}) = 0.$

Therefore we obtain the first integral of motion of the Korteweg-de Vries equation in the form

$\int_{- \infty}^{+ \infty} u d x = c o n s t .$

In order to look for the other integrals of motion Gardner suggested to use the following Miura transform

$u (x, t) = w - ε w_{x} - ε^{2} w^{2}, w = w (x, t) .$

One obtains the following correlation

$u_{t} + 6 u u_{x} + u_{x x x} = (1 - 2 ε^{2} w - ε \frac{\partial}{\partial x}) [w_{t} + 6 (w - ε^{2} w^{2}) w {}_{x}+ w_{x x x}] .$

One can notice that

u (x, t)

doesn't contain parameter

ε .

It is possible if

w = w (x, t, ε)

. Let us look for

w (x, t, ε)

in the form

$w (x, t, ε) = \sum_{n = 0}^{\infty} w_{n} (x, t) ε^{n} .$

From the equation

$w_{t} + 6 (w - ε^{2} w^{2}) w_{x} + w_{x x x} = 0$

we get

$\int_{- \infty}^{+ \infty} w (x, t, ε) d x = c o n s t .$

Therefore we get the following set of relations

$\int_{- \infty}^{+ \infty} w_{n} (x, t) d x = c o n s t, n = 1,2, \dots$

Using the Miura transform to return to function

u (x, t)

we obtain

$w_{0} = u,$

$w_{1} = w_{0, x} = u_{x},$

$w_{2} = w_{1, x} + w_{0}^{2} = u_{x x} + u^{2},$

$w_{3} = w_{2, x} + 2 w_{0} w_{1} = u_{x x x} + 4 u u_{x},$

$w_{4} = w_{3, x} + 2 w_{0} w_{2} + w_{1}^{2} = u_{x x x x} + 6 u u_{x x} + 5 u_{x}^{2} + 2 u^{3} .$

All these expressions give integrals of motion for the Korteweg-de Vries equation. One can continue the search procedure and obtain an infinite number of integrals of motion. This important property is typical for equations solvable by the inverse scattering transform.

Miura transform and Lax pair

Miura found out that transform (now known as Miura transform)

$u = v_{x} - v^{2}$

lets one express solutions of the Korteweg-de Vries equation

$u_{t} + 6 u u_{x} + u_{x x x} = 0$

via known solutions of the modified Korteweg-de Vries equation

$v_{t} - 6 v^{2} v_{x} + v_{x x x} = 0.$

Details

One can notice that we can introduce an arbitrary constant

λ

into Miura transform so it takes the form

$v_{x} = u + λ + v^{2} .$

Now, one can rewrite the modified Korteweg-de Vries equation in the following way

$\begin{array}{l} v_{t} - 6 v^{2} v_{x} + v_{x x x} = v_{t} + \frac{\partial}{\partial x} [(\frac{\partial}{\partial x} + 2 v) (v_{x} - v^{2})] = \\ = v_{t} + \frac{\partial}{\partial x} [(\frac{\partial}{\partial x} + 2 v) u] . \end{array}$

So we can treat Miura transform and the modified Korteweg-de Vries equation as a system of equations

$v_{x} = u + λ + v^{2}$

$v_{t} = - \frac{\partial}{\partial x} [(\frac{\partial}{\partial x} + 2 v) (u - 2 λ)]$

with compatibility condition

${(v_{x})}_{t} = {(v_{t})}_{x} .$

Let us introduce new unknown function

ψ (x, t)

$v = - \frac{ψ_{x}}{ψ} .$

Then we get the following system of linear partial differential equations

$ψ_{x x} + (u + λ) ψ = 0$

$ψ_{t} = (c (t) + u_{x}) ψ - 2 (u - 2 λ) ψ_{x} .$

Obtained system can be written in the form

$\hat{L} ψ = λ ψ$

$ψ_{t} = \hat{A} ψ,$

where

\hat{L}

is the Schrödinger operator

$\hat{L} = - \frac{\partial^{2}}{\partial x^{2}} - u$

and

\hat{A}

is an auxiliary evolutionary operator given by

$\hat{A} = 2 (2 λ - u) \frac{\partial}{\partial x} + (c (t) + u_{x}) .$

Using compatibility condition

${(ψ_{x x})}_{t} = {(ψ_{t})}_{x x}$

we find that the Korteweg-de Vries equation is equivalent to operator equation

${\hat{L}}_{t} + [\hat{L}, \hat{A}] = 0,$

where

[\hat{L}, \hat{A}]

is the commutator defined as

$[\hat{L}, \hat{A}] = \hat{L} \hat{A} - \hat{A} \hat{L} .$

The Korteweg-de Vries equation is a nonlinear partial differential equation that can not be reduced to one linear equation. However it is equivalent to the system of linear equations. Using this system one can construct solution of the Cauchy problem for the Korteweg-de Vries equation using the inverse scattering transform.

The Cauchy problem

The Cauchy problem for the Korteweg-de Vries equation can be solved using the inverse scattering transform (IST). This method was first discovered by Gardner, Greene, Kruskal and Miura in 1967 (Gardner, et al., 1967).

See Application of the inverse scattering transform for the Korteweg-de Vries equation.

Lagrangian

The Korteweg-de Vries equation

$u_{t} + 6 u u_{x} + u_{x x x} = 0$

can be written in form

$\frac{\partial L}{\partial u} - \frac{\partial}{\partial x} \frac{\partial L}{\partial u_{x}} - \frac{\partial}{\partial t} \frac{\partial L}{\partial u_{t}} = 0$

where

L

is a Lagrangian

$L = \frac{1}{2} ψ_{x} ψ_{t} + ψ_{x}^{3} - \frac{1}{2} ψ_{x x}^{2}$

with

u

defined by

$u = ψ_{x} .$

Hamiltonian

The Korteweg-de Vries equation

$u_{t} + 6 u u_{x} + u_{x x x} = 0$

can be written in form

$u_{t} = \frac{\partial}{\partial x} \frac{δ H}{δ u},$

where

\frac{δ}{δ u}

is a variational derivative given by

$\lim_{ε \to 0} \frac{1}{ε} (H [u + ε δ u] - H [u]) = \int_{- \infty}^{+ \infty} \frac{δ H}{δ u} δ u d x$

and

H

is a Hamiltonian function

$H = \int_{- \infty}^{\infty} (\frac{1}{2} u_{x}^{2} - u^{3}) d x .$

Thus we see that the Korteweg-de Vries equation can be presented as an infinite dimensional Hamiltonian system.

Applications and connections

The Korteweg-de Vries equation is used as a model in fields as wide as:

acoustic waves on a crystal lattice
ion-acoustic waves in a plasma
long internal waves in a density-stratified ocean
shallow-water waves with weakly non-linear restoring forces

and more.

Variations

Modified Korteweg-de Vries equation

$u_{t} - 6 u^{2} u_{x} + u_{x x x} = 0$

Generalized Korteweg-de Vries equation

$u_{t} + f (u) u_{x} + u_{x x x} = 0$

Cylindrical Korteweg-de Vries equation

$u_{t} - 6 u u_{x} + u_{x x x} + \frac{u}{2 t} = 0$

References

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Boussinesq, J. Essai sur la theorie des eaux courantes, Memoires presentes par divers savants l’Acad. des Sci. Inst. Nat. France, 1877. XXIII. Pp. 1–680.
Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C. Solitons and nonlinear wave equations // Academic Press, New York, 1982. — 630 p.
Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M. Method for solving the Korteweg-de Vries equation // Phys. Rev. Lett., 1967. 19:19. Pp.1095-1097. DOI:10.1103/PhysRevLett.19.1095
Gel'fand I.M., Levitan B.M. On the determination of a differential equation from its spectral function (in Russian) // Izv. Akad. Nauk SSSR Ser. Mat., 1951. 15:4. Pp.309-360.
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Hirota R. The direct method in soliton theory (translated from the Japanese and edited by Nagai A., Nimmo J. and Gilson C.) // Cambridge Tracts in Mathematics, 2004. 155. — 200 p.
Korteweg D.J., de Vries G. On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary wave // Phil. Mag. Series 5, 1895. 39:240. Pp.422-443. DOI:10.1080/14786449508620739
Kudryashov N.A. Analytical theory of nonlinear differential equations (in Russian) // 2^nd ed., Institute of Computer Investigation, Moscow-Izhevsk, 2004. — 360 p.
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External links

Korteweg-de Vries equation in Encyclopaedia of Mathematics
Korteweg-de Vries equation in EqWorld, the world of mathematical equations
Korteweg-de Vries equation in Wolfram MathWorld
Korteweg-de Vries equation in Scholarpedia
Korteweg-de Vries equation in Wikipedia, the free encyclopedia
The Korteweg-de Vries equation: history, exact solutions, and graphical representation by Klaus Brauer

Categories: Equation

From Neqwiki, the nonlinear equations encyclopedia

Korteweg-de Vries equation (KdV)

Contents

Definition

History

Solutions

Traveling wave

Soliton

N-soliton solutions

Self-similar solutions

Integrals of motion

Miura transform and Lax pair

The Cauchy problem

Lagrangian

Hamiltonian

Applications and connections

Variations

See also

References

External links

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