# Korteweg-de Vries equation

KdV-equation

The equation

$$\frac{\partial u }{\partial t } - 6u \frac{\partial u }{\partial x } + \frac{\partial ^ {3} u }{\partial x ^ {3} } = 0; \ \ x, t \in \mathbf R ^ {1} ,\ \ u ( x, t) \in \mathbf R ^ {1} .$$

It was proposed by D. Korteweg and G. de Vries  to describe wave propagation on the surface of shallow water. It can be interpreted using the inverse-scattering method, which is based on presenting the KdV-equation in the form

$$\frac{\partial L }{\partial t } = [ L, M] = LM - ML,$$

where $L = - {\partial ^ {2} } / {\partial x ^ {2} } + u ( x, t)$ is the one-dimensional Schrödinger operator and

$$M = 4 \frac{\partial ^ {3} }{\partial x ^ {3} } - 3 \left ( u \frac \partial {\partial x } + \frac \partial {\partial x } u \right ) .$$

The Cauchy problem for the KdV-equation is uniquely solvable in the class of rapidly decreasing functions $u$ with the initial condition: $u \in S ( \mathbf R ^ {1} )$ (where $S ( \mathbf R ^ {1} )$ is the Schwartz space). Let

$$s = \{ r ( k) \in S ( \mathbf R ^ {1} ) :\ \kappa _ {j} , m _ {j} > 0,\ \kappa _ {j _ {1} } \neq \kappa _ {j _ {2} } \ \textrm{ if } \ j _ {1} \neq j _ {2} ,$$

$$j = 1 \dots n {} \}$$

be the scattering data for the Schrödinger operator with potential $u ( x)$, $\{ x _ {j} \}$ the discrete spectrum, $\{ k \}$ the continuous spectrum and $m _ {j}$ normalization coefficients of the eigen functions. Then

$$s ( t) = \ \{ r ( k, t) = \ e ^ {8ik ^ {3} t } r ( k),\ \ m _ {j} ( t) = \ e ^ {8 \kappa _ {j} ^ {3} t } m _ {j} ,\ \ \kappa _ {j} ( t) = \ \kappa _ {j} \}$$

are the scattering data for the Schrödinger equation with potential $u ( x, t)$, and the solution $u ( x, t)$ is determined from the scattering data $s ( t)$ using a certain integral equation. If $r ( k) = 0$, the latter equation can be solved explicitly; the potentials thus obtained are known as reflection-free, and the corresponding solutions of the KdV-equation are known as $n$-solitons (see Soliton).

The KdV-equation may be written in Hamiltonian form

$$\frac{\partial u }{\partial t } = \ \frac \partial {\partial x } \frac{\delta H }{\delta u } ,\ \ H ( u) = \ \int\limits _ {- \infty } ^ { {+ } \infty } \left ( u ^ {3} + { \frac{1}{2} } \left ( \frac{\partial u }{\partial x } \right ) ^ {2} \right ) dx;$$

here the phase space is $S ( \mathbf R ^ {1} )$ and the Poisson brackets are defined by the bilinear form of the operator $\partial / {\partial x }$. The mapping $u ( x) \rightarrow s$ is a canonical transformation to variables of action-angle type. In terms of these new variables the Hamilton equations can be integrated explicitly. The KdV-equation possesses infinitely many integrals of motion:

$$I _ {n} ( u) = \ \int\limits _ {- \infty } ^ { {+ } \infty } P _ {2n - 1 } \left ( u ,\ \frac{\partial u }{\partial x } \dots \frac{\partial ^ {n - 2 } u }{\partial x ^ {n - 2 } } \right ) dx;$$

$$P _ {1} = u ,\ P _ {n} = - \frac{\partial P _ {n - 1 } }{dx } + \sum _ {j = 1 } ^ { {n } - 2 } P _ {n - 1 - j } P _ {j} ,\ n > 1.$$

All these integrals of motion are integrals in involution, and the Hamiltonian systems that they generate (known as higher Korteweg–de Vries equations) are completely integrable.

Using the integral equations of the inverse problem, one can also find the solution of the Cauchy problem for step-type initial data:

$$\lim\limits _ {x \rightarrow - \infty } u ( x) = c < 0,\ \ \lim\limits _ {x \rightarrow + \infty } u ( x) = 0.$$

As $t \rightarrow + \infty$ in a neighbourhood of the front, the solution $u ( x, t)$ decomposes into non-interacting solitons — this is the process of step disintegration.

In the case of the Cauchy problem with periodic initial data $u ( x + T) = u ( x)$, $x \in \mathbf R ^ {1}$, the analogue of reflection-free potentials is provided by potentials for which the Schrödinger operator has finitely many forbidden zones — finite-gap potentials. Periodic and almost-periodic finite-gap potentials are stationary solutions of the higher KdV-equations; the latter constitute completely-integrable finite-dimensional Hamiltonian systems. Any periodic potential can be approximated by a finite-gap potential. Let $E _ {j} \in \mathbf R ^ {1}$, $E _ {j _ {1} } \neq E _ {j _ {2} }$ if $j _ {1} \neq j _ {2}$ ($j = 1 \dots 2g + 1$), be the edges of the bands, and $\Gamma$ the hyperelliptic curve

$$y ^ {2} = \ \prod _ {j = 1 } ^ { {2g } + 1 } ( x - E _ {j} )$$

over the field $\mathbf C$. Then real-valued almost-periodic potentials with the above band edges, as well as solutions of the Cauchy problem, are expressible in terms of $\theta$-functions on the Jacobi variety $J ( \Gamma )$ of the curve $\Gamma$. Subject to certain conditions on the edges, the resulting solutions will be periodic. If one drops the conditions $E _ {j} \in \mathbf R ^ {1}$, one obtains complex-valued solutions of the KdV-equation (possibly with poles), which are also called finite-gap potentials.

How to Cite This Entry:
Korteweg-de Vries equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Korteweg-de_Vries_equation&oldid=51990
This article was adapted from an original article by L.A. Takhtadzhyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article