Korteweg-de Vries equation
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H[u] = \int u_x^2 + u^3 dx} with symplectic phase space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{-1/2}} shared by other KdV-type equations. F. Magri has shown Mag78 that KdV may also be represented using the Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H[u] = \int u^2 dx} . The natural phase space associated to the Magri representation of KdV appears to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{-3/2}} but details need to be worked out.
Symplectic nonsqueezing of the KdV flow in the associated symplectic phase space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H_0^{-1/2} (\mathbb{T})} was established in CoKeStTkTa2004. Whether nonsqueezing also holds in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{-3/2}} using the Magri representation is unknown. The periodic KdV flow is not known to be globally well-posed in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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)
