Korteweg-de Vries equation

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The Korteweg-de Vries (KdV) equation is

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 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 with symplectic phase space shared by other KdV-type equations. F. Magri has shown Mag78 that KdV may also be represented using the Hamiltonian . The natural phase space associated to the Magri representation of KdV appears to be but details need to be worked out.

Symplectic nonsqueezing of the KdV flow in the associated symplectic phase space was established in CoKeStTkTa2004. Whether nonsqueezing also holds in using the Magri representation is unknown. The periodic KdV flow is not known to be globally well-posed in .

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