# Korteweg-de Vries equation

The Korteweg-de Vries (KdV) equation is

${\displaystyle \partial _{t}u+\partial _{x}^{3}u+6u\partial _{x}u=0.}$

The factor of 6 is convenient for reasons of complete integrability, but can easily be scaled out if desired.

The equation is completely integrable, and has infinitely many conserved quantities. Indeed, for each non-negative integer k, there is a conserved quantity which is roughly equivalent to the ${\displaystyle H^{k}}$ norm of u.

The KdV equation has been studied on the line, on the circle, and on the half-line.

The KdV equation is the first non-trivial equation on the KdV hierarchy and is the most famous member of the family of KdV-type equations.

## Symplectic Structures

At least two distinct Hamiltonian representations of the completely integrable Korteweg-de Vries equation are known. The standard Fadeev-Zakharov representation uses Hamiltonian ${\displaystyle H[u]=\int u_{x}^{2}+u^{3}dx}$ with symplectic phase space ${\displaystyle H^{-1/2}}$ shared by other KdV-type equations. F. Magri has shown Mag78 that KdV may also be represented using the Hamiltonian ${\displaystyle H[u]=\int u^{2}dx}$. The natural phase space associated to the Magri representation of KdV appears to be ${\displaystyle H^{-3/2}}$ but details need to be worked out.

Symplectic nonsqueezing of the KdV flow in the associated symplectic phase space ${\displaystyle H_{0}^{-1/2}(\mathbb {T} )}$ was established in CoKeStTkTa2004. Whether nonsqueezing also holds in ${\displaystyle H^{-3/2}}$ using the Magri representation is unknown. The periodic KdV flow is not known to be globally well-posed in ${\displaystyle H^{-3/2}}$.

Question: Do there exist other symplectic representations of KdV besides the Fadeev-Zakharov and Magri representations? Colliand 12:01, 14 September 2006 (EDT)