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SUMMARY:A completeness study on a class of Lax pairs - Hay\, M (Kyushu)
DTSTART:20090702T103000Z
DTEND:20090702T110000Z
UID:TALK18970@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Despite the existence of a Lax pair for a given equation often
  being used as the definition of its integrability\, recently there have b
 een few studies that seek to find or categorize nonlinear equations that u
 sing Lax pairs as their starting point. Of those studies that do begin wit
 h Lax pairs\, all choose a priori a form of the Lax pair\, i.e. the type o
 f dependence on the spectral parameter\, thus limiting the possible result
 s. \n\nWe present a study that begins with Lax pairs for partial differenc
 e equations that are 2x2\, where each entry of the Lax matrices contains o
 nly one separable term. Besides that the terms are general\, no ansatz is 
 made about the dependence on the spectral parameter\, nor about the terms 
 dependent on the lattice variables. We examine all possible groups of equa
 tions that can arise via the compatibility condition. Once a system of equ
 ations is extracted from the compatibility condition\, it is solved in a w
 ay that preserves its generality\, up to a point where nonlinear evolution
  equations are apparent\, or the system is shown to be trivial. \n\nIn fac
 t\, of all the potential Lax pairs identified by this method\, only two le
 ad to non-trivial and unconstrained evolution equations. These are new\, h
 igher order varieties of the lattice sine-Gordon (LSG) and the lattice mod
 ified KdV (LMKDV) equations. The remaining systems are shown to be trivial
 \, overdetermined or underdetermined\, which suggests a connection between
  the existence of a Lax pair and the singularity confinement method of ide
 ntifying integrable equations. \n\nAs we do not make any assumptions about
  the explicit dependence on the spectral parameter\, we show that a partic
 ular nonlinear equation may have many Lax pairs\, all depending on the spe
 ctral parameter in different ways. The effect that this freedom has on the
  process of inverse scattering is\, as yet\, unclear.
LOCATION:Seminar Room 1\, Newton Institute
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