Korteweg-de Vries equation: Difference between revisions
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The '''Korteweg-de Vries (KdV) equation''' is | The '''Korteweg-de Vries (KdV) equation''' is | ||
<center><math> | <center><math>\partial_t u + \partial_x^3 u + 6u\partial_x u = 0.</math></center> | ||
The factor of 6 is convenient for reasons of [[complete integrability]], but can easily be scaled out if desired. | The factor of 6 is convenient for reasons of [[completely integrable|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 < | 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 <math>H^k</math> norm of u. | ||
The <span class="SpellE">KdV</span> equation has been studied [[Korteweg-de Vries equation on R|on the line]], [[Korteweg-de Vries equation on T|on the circle]], and [[Korteweg-de Vries equation on the half-line|on the half-line]]. | The <span class="SpellE">KdV</span> equation has been studied [[Korteweg-de Vries equation on R|on the line]], [[Korteweg-de Vries equation on T|on the circle]], and [[Korteweg-de Vries equation on the half-line|on the half-line]]. | ||
The KdV equation is the first non-trivial equation on the [[KdV hierarchy]]. | 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| completely integrable]] [[Korteweg-de Vries equation]] are known. The standard Fadeev-Zakharov representation uses Hamiltonian <math>H[u] = \int u_x^2 + u^3 dx</math> with symplectic phase space <math>H^{-1/2}</math> shared by other [[KdV-type equations]]. F. Magri has shown [[Mag78]] that KdV may also be represented using the Hamiltonian <math> H[u] = \int u^2 dx</math>. The natural phase space associated to the Magri representation of KdV appears to be <math>H^{-3/2}</math> but details need to be worked out. | |||
[[Symplectic nonsqueezing]] of the KdV flow in the associated symplectic phase space <math>H_0^{-1/2} (\mathbb{T})</math> was established in [[CoKeStTkTa2004]]. Whether nonsqueezing also holds in <math>H^{-3/2}</math> using the Magri representation is unknown. The periodic KdV flow is not known to be globally well-posed in <math>H^{-3/2}</math>. | |||
:Question: Do there exist other symplectic representations of KdV besides the Fadeev-Zakharov and Magri representations? [[User:Colliand|Colliand]] 12:01, 14 September 2006 (EDT) | |||
[[Category:Integrability]] | |||
[[Category:Equations]] | [[Category:Equations]] | ||
[[Category:Airy]] | |||
Latest revision as of 16:10, 14 September 2006
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)
