BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:A Special Collection of Traveling Waves for the Kawahara Equation 
 - Michael Shearer (North Carolina State University)
DTSTART:20220906T143000Z
DTEND:20220906T153000Z
UID:TALK177818@talks.cam.ac.uk
DESCRIPTION:The Kawahara equation is a 5th order dispersive equation\, a h
 igher order version of the KdV equation. Traveling waves (TWs) satisfy a f
 ourth-order ordinary differential equation in which the traveling wave spe
 ed c and a constant of integration A are parameters. A further integration
  yields the Hamiltonian\, an invariant of all solutions. Periodic solution
 s are computed with an iterative spectral method\, resulting in a family o
 f periodic solutions depending on the three constants c\, A and wave numbe
 r k. We derive jump conditions between periodic solutions with different w
 ave numbers but equal speeds and Hamiltonian. The jump conditions are nece
 ssary conditions for the existence of traveling waves that asymptote to th
 e periodic orbits at infinity. Bifurcation theory and parameter continuati
 on are then used to compute multiple solution branches for the jump condit
 ions. From these\, we construct heteroclinic orbits from the intersection 
 of stable and unstable manifolds of compatible periodic solutions. Each br
 anch terminates at an equilibrium-to-periodic solution in which the equili
 brium is the background for a solitary wave that connects to the associate
 d periodic solution. The jump conditions are closely related to Whitham sh
 ocks\, discontinuous solutions of the Whitham modulation equations\, sugge
 sting the existence of wave structures more complex than the traveling wav
 es presented here.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR