# Korteweg-de Vries equation (KdV)

Korteweg-de Vries equation
Order3rd
SolvableExactly solvable
The Cauchy problemSolvable by inverse scattering transform
HierarchyKorteweg-de Vries hierarchy
First introducedKorteweg & de Vries, 1895

## Definition

${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$

## 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 19th century. This solution can be expressed via Jacobi's elliptic functions. Let us look for solutions of the Korteweg-de Vries equation

${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$

in the form of traveling wave

$u\left(x,t\right)=y\left(z\right),\text{ }z=x-{c}_{0}t.$

Then the Korteweg-de Vries equation will take the form

${y}_{zz}+3{y}^{2}-{c}_{0}y+{c}_{1}=0.$

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

$u\left(x,t\right)=\beta +\left(\alpha -\beta \right)c{n}^{2}\left\{\sqrt{\frac{\alpha -\gamma }{2}}\left(z-{z}_{0}\right),{S}^{2}\right\}$

with $0\le S\le 1$.

This solution is a wave with speed ${c}_{0}$ and period $T$ defined by the following expressions

${c}_{0}=2\left(\alpha +\beta +\gamma \right),$

$T=2\sqrt{\frac{2}{\alpha -\gamma }}\underset{0}{\overset{1}{\int }}\frac{dx}{\sqrt{\left(1-{x}^{2}\right)\left(1-{S}^{2}{x}^{2}\right)}}=\sqrt{\frac{8}{\alpha -\gamma }}K\left(S\right),$

where $K\left(S\right)$ 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 $\alpha >\beta =\gamma$ then ${S}^{2}=1$ and the period $T$ tends to infinity and we obtain solitary wave

$u\left(x,t\right)=y\left(z\right)=\beta +\left(\alpha -\beta \right)c{h}^{-2}\left\{\sqrt{\frac{\alpha -\beta }{2}}\left(z-{z}_{0}\right)\right\}.$

Usually one sets $\beta =\gamma =0$, $\alpha =2{k}^{2}$ so the solution takes the form

$u\left(x,t\right)=2{k}^{2}c{h}^{-2}\left\{k\left(x-4{k}^{2}t\right)+{\chi }_{0}\right\},$

where ${\chi }_{0}$ 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

${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$

in the form

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

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

${w}_{zz}-2{w}^{3}-zw+\alpha =0.$

This is the second Painlevé equation. Solutions of this equation generally can be expressed via the so-called Painlevé transcendents. For integer $\alpha$ the second Painlevé equation possesses rational solutions; for half-integer $\alpha$ 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

${u}_{t}+6u{u}_{x}+{u}_{xxx}=0.$

Using the Miura transform to return to function $u\left(x,t\right)$ we obtain

${w}_{0}=u,$

${w}_{1}={w}_{0,x}={u}_{x},$

${w}_{2}={w}_{1,x}+{w}_{0}^{2}={u}_{xx}+{u}^{2},$

${w}_{3}={w}_{2,x}+2{w}_{0}{w}_{1}={u}_{xxx}+4u{u}_{x},$

${w}_{4}={w}_{3,x}+2{w}_{0}{w}_{2}+{w}_{1}^{2}={u}_{xxxx}+6u{u}_{xx}+5{u}_{x}^{2}+2{u}^{3}.$

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

${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$

via known solutions of the modified Korteweg-de Vries equation

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

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

${\stackrel{^}{L}}_{t}+\left[\stackrel{^}{L},\stackrel{^}{A}\right]=0,$

where $\left[\stackrel{^}{L},\stackrel{^}{A}\right]$ is the commutator defined as

$\left[\stackrel{^}{L},\stackrel{^}{A}\right]=\stackrel{^}{L}\stackrel{^}{A}-\stackrel{^}{A}\stackrel{^}{L}.$

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

${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$

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

$L=\frac{1}{2}{\psi }_{x}{\psi }_{t}+{\psi }_{x}^{3}-\frac{1}{2}{\psi }_{xx}^{2}$

with $u$ defined by

$u={\psi }_{x}.$

## Hamiltonian

The Korteweg-de Vries equation

${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$

can be written in form

${u}_{t}=\frac{\partial }{\partial x}\frac{\delta H}{\delta u},$

where $\frac{\delta }{\delta u}$ is a variational derivative given by

$\underset{\epsilon \to 0}{\mathrm{lim}}\frac{1}{\epsilon }\left(H\left[u+\epsilon \delta u\right]-H\left[u\right]\right)=\underset{-\infty }{\overset{+\infty }{\int }}\frac{\delta H}{\delta u}\delta udx$

and $H$ is a Hamiltonian function

$H=\underset{-\infty }{\overset{\infty }{\int }}\left(\frac{1}{2}{u}_{x}^{2}-{u}^{3}\right)dx.$

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

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

Generalized Korteweg-de Vries equation

${u}_{t}+f\left(u\right){u}_{x}+{u}_{xxx}=0$

Cylindrical Korteweg-de Vries equation

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

Korteweg-de Vries hierarchy

Modified Korteweg-de Vries equation

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