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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:A Special Collection of Traveling Waves for the Ka
wahara Equation - Michael Shearer (North Carolina
State University)
DTSTART;TZID=Europe/London:20220906T153000
DTEND;TZID=Europe/London:20220906T163000
UID:TALK177818AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/177818
DESCRIPTION:The Kawahara equation is a 5th order dispersive eq
uation\, a higher order version of the KdV equatio
n. Traveling waves (TWs) satisfy a fourth-order or
dinary differential equation in which the travelin
g wave speed c and a constant of integration A are
parameters. A further integration yields the Hami
ltonian\, an invariant of all solutions. Periodic
solutions are computed with an iterative spectral
method\, resulting in a family of periodic solutio
ns depending on the three constants c\, A and wave
number k. We derive jump conditions between perio
dic solutions with different wave numbers but equa
l speeds and Hamiltonian. The jump conditions are
necessary conditions for the existence of travelin
g waves that asymptote to the periodic orbits at i
nfinity. Bifurcation theory and parameter continua
tion are then used to compute multiple solution br
anches for the jump conditions. From these\, we co
nstruct heteroclinic orbits from the intersection
of stable and unstable manifolds of compatible per
iodic solutions. Each branch terminates at an equi
librium-to-periodic solution in which the equilibr
ium is the background for a solitary wave that con
nects to the associated periodic solution. The jum
p conditions are closely related to Whitham shocks
\, discontinuous solutions of the Whitham modulati
on equations\, suggesting the existence of wave st
ructures more complex than the traveling waves pre
sented here. \;
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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