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>u_t + u_x^3 + 6uu_x = 0.</math></center>
<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 <span class="SpellE">H^k</span> norm of u.
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 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)