Redirected from Korteweg-De Vries Hierarchy

# Korteweg-de Vries hierarchy (KdV hierarchy)

Korteweg-de Vries hierarchy
SolvableExactly solvable
The Cauchy problemSolvable by inverse scattering transform
First introducedLax, 1968

## Definition

The Korteweg-de Vries equation is the first equation of the so-called Korteweg-de Vries hierarchy defined as

${u}_{t}+\frac{\partial }{\partial x}{L}_{n+1}\left[u\right]=0,$

where $L$ is Lenard operator defined by

$\frac{\partial }{\partial x}{L}_{n+1}\left[u\right]=\left(\frac{{\partial }^{3}}{\partial {x}^{3}}+4u\frac{\partial }{\partial x}+2\frac{\partial u}{\partial x}\right){L}_{n}\left[u\right],\text{ }{L}_{0}\left[u\right]=\frac{1}{2}.$

There are four first operators

${L}_{1}\left[u\right]=u,$

${L}_{2}\left[u\right]={u}_{xx}+3{u}^{2},$

${L}_{3}\left[u\right]={u}_{xxxx}+10u{u}_{xx}+5{u}_{x}^{2}+10{u}^{3},$

${L}_{4}\left[u\right]={u}_{xxxxxx}+14u{u}_{xxxx}+28{u}_{x}{u}_{xxx}+21{u}_{xx}^{2}+70{u}^{2}{u}_{xx}+70u{u}_{x}^{2}+35{u}^{4}.$

Corresponding equations of the Korteweg-de Vries hierarchy are

${u}_{t}+{u}_{x}=0,$

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

${u}_{t}+10u{u}_{xxx}+30{u}^{2}{u}_{x}+20{u}_{x}{u}_{xx}+{u}_{5x}=0,$

$\begin{array}{c}{u}_{t}+14u{u}_{5x}+70{u}^{2}{u}_{xxx}+42{u}_{x}{u}_{4x}+\\ +70{u}_{xx}{u}_{xxx}+280u{u}_{x}{u}_{xx}+70{u}_{x}^{3}+140{u}^{3}{u}_{x}+{u}_{7x}=0.\end{array}$

All equations of the Korteweg-de Vries hierarchy pass the Painlevé test, possess soliton solutions and are exactly solvable. Also all equations has odd degrees that corresponds to the lack of dissipative terms.

## Derivation

In 1968 P.D. Lax solved problem of finding all partial differential equations for which the Cauchy problem can be solved by the inverse scattering transform with Schrödinger steady-state equation. He suggested to look for equation class representable as a system of two linear equations

${\psi }_{xx}+\left(\lambda +u\right)\psi =0$

${\psi }_{t}=B\left(u,{u}_{x},\dots ,\lambda \right)\psi -2E\left(u,{u}_{x},\dots ,\lambda \right){\psi }_{x},$

where $B$ and $E$ are functions of $\lambda$, $u$ and derivatives form $u$ by $x$ which are to be found.

## Lax pair

Lax pair for the nth equation of the Korteweg-de Vries hierarchy can be written as the following system of linear partial differential equations

${\psi }_{xx}+\left(\lambda +u\right)\psi =0,$

${\psi }_{t}=\left(c\left(t\right)+\sum _{k=0}^{n}{\left(-4\lambda \right)}^{n-k}{L}_{x}^{k}\left[u\right]\right)\psi -2\left(\sum _{k=0}^{n}{\left(-4\lambda \right)}^{n-k}{L}^{k}\left[u\right]\right){\psi }_{x}.$

Korteweg-de Vries equation

Modified Korteweg-de Vries hierarchy

## References

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