Korteweg-de Vries equation

The Korteweg-de Vries (KdV) equation is

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

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 $H^{k}$ 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 $H[u]=\int u_{x}^{2}+u^{3}dx$ with symplectic phase space $H^{-1/2}$ shared by other KdV-type equations. F. Magri has shown Mag78 that KdV may also be represented using the Hamiltonian $H[u]=\int u^{2}dx$. The natural phase space associated to the Magri representation of KdV appears to be $H^{-3/2}$ but details need to be worked out.

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

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