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CATEGORIES:Applied and Computational Analysis
SUMMARY:Singular solutions of a modified two-component sha
llow-water equation - Darryl Holm (Imperial Colleg
e)
DTSTART;TZID=Europe/London:20081023T150000
DTEND;TZID=Europe/London:20081023T160000
UID:TALK13530AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/13530
DESCRIPTION:The Camassa-Holm equation (CH) is a well known int
egrable Hamiltonian equation describing the veloci
ty dynamics of shallow water waves. In its dispers
ionless limit\, this equation exhibits spontaneous
emergence of singular solutions (peakons) from sm
ooth initial conditions. The CH equation has been
recently extended to a two-component integrable sy
stem (CH2)\, which includes both velocity and dens
ity variables in the dynamics. Although 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 depe
ndence on average density as well as on pointwise
density. The modified CH2 system (MCH2) now admits
peakon solutions in both velocity and average den
sity\, although it may no longer be integrable. We
analytically identify the steepening mechanism th
at summons the emergent singular solutions from sm
ooth spatially-confined initial data.\n\nNumerical
results for MCH2 are given and compared with the
pure CH2 case. These numerics show that the modifi
cation in MCH2 to introduce average density has li
ttle short-time effect on the emergent dynamical p
roperties. However\, an analytical and numerical s
tudy of pairwise peakon interactions for MCH2 show
s a new asymptotic feature. Namely\, besides the e
xpected soliton scattering behavior seen in both o
vertaking and head-on peakon collisions\, MCH2 als
o allows the phase shift of the peakon collision t
o diverge in certain parameter regimes.
LOCATION:MR14\, CMS
CONTACT:
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