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SUMMARY:Singular solutions of a modified two-component shallow-water equat
 ion - Darryl Holm (Imperial College)
DTSTART:20081023T140000Z
DTEND:20081023T150000Z
UID:TALK13530@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:The Camassa-Holm equation (CH) is a well known integrable Hami
 ltonian equation describing the velocity dynamics of shallow water waves. 
 In its dispersionless limit\, this equation exhibits spontaneous emergence
  of singular solutions (peakons) from smooth initial conditions. The CH eq
 uation has been recently extended to a two-component integrable system (CH
 2)\, which includes both velocity and density variables in the dynamics. A
 lthough possessing peakon solutions in the velocity\, the CH2 system does 
 not admit singular solutions in its density profile.\n\nWe modify the CH2 
 system to allow dependence on average density as well as on pointwise dens
 ity. The modified CH2 system (MCH2) now admits peakon solutions in both ve
 locity and average density\, although it may no longer be integrable. We a
 nalytically identify the steepening mechanism that summons the emergent si
 ngular solutions from smooth spatially-confined initial data.\n\nNumerical
  results for MCH2 are given and compared with the pure CH2 case. These num
 erics show that the modification in MCH2 to introduce average density has 
 little short-time effect on the emergent dynamical properties. However\, a
 n analytical and numerical study of pairwise peakon interactions for MCH2 
 shows a new asymptotic feature. Namely\, besides the expected soliton scat
 tering behavior seen in both overtaking and head-on peakon collisions\, MC
 H2 also allows the phase shift of the peakon collision to diverge in certa
 in parameter regimes.
LOCATION:MR14\, CMS
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