Korteweg-de Vries equation
From DispersiveWiki
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 Hk 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](/wiki/images/math/0/9/9/099ed90c8f9dfe834fbcd457312948ef.png)
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](/wiki/images/math/4/5/d/45d0b294487901ec1355e6e12d9ca56a.png)
. 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 
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)
