Diff (S1) and Adler-Gelfand-Dikii spaces and integrable systems

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We study a family of Korteweg-deVries equations as some evolution equations associated to the Adler-Gelfand-Dikii (AGD) space. First we derive (formally) the Korteweg-deVries (KdV) as an evolution equation of the AGD operator under the action of Vect(S1).The solutions of the AGD operator define an immersion C → ℂPn-1 in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated to Δ(n)= dn/dxn+un-2 dn-2/dxn-2 + ... + u0. This equation is invariant under linear fractional transformations. We also show how the modified KdV is related to the Schwarzian KdV by the Cole-Hopf transformation. The geometrical (differential Galois theory) connections between all these equations are given.

Original languageBritish English
Pages (from-to)17-31
Number of pages15
JournalLetters in Mathematical Physics
Issue number1
StatePublished - Jan 2001


  • AGD space
  • Galois theory
  • Integrable systems
  • KdV equations


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