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>\partial_t u + \partial_x^3 u + 6u\ | <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 [[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]]. | ||
Revision as of 19:12, 28 July 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.
