A Special Collection of Traveling Waves for the Kawahara Equation

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• Michael Shearer (North Carolina State University)
• Tuesday 06 September 2022, 15:30-16:30
• Seminar Room 1, Newton Institute.

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HY2W02 - Analysis of dispersive systems

The Kawahara equation is a 5th order dispersive equation, a higher order version of the KdV equation. Traveling waves (TWs) satisfy a fourth-order ordinary differential equation in which the traveling wave speed c and a constant of integration A are parameters. A further integration yields the Hamiltonian, an invariant of all solutions. Periodic solutions are computed with an iterative spectral method, resulting in a family of periodic solutions depending on the three constants c, A and wave number k. We derive jump conditions between periodic solutions with different wave numbers but equal speeds and Hamiltonian. The jump conditions are necessary conditions for the existence of traveling waves that asymptote to the periodic orbits at infinity. Bifurcation theory and parameter continuation are then used to compute multiple solution branches for the jump conditions. From these, we construct heteroclinic orbits from the intersection of stable and unstable manifolds of compatible periodic solutions. Each branch terminates at an equilibrium-to-periodic solution in which the equilibrium is the background for a solitary wave that connects to the associated periodic solution. The jump conditions are closely related to Whitham shocks, discontinuous solutions of the Whitham modulation equations, suggesting the existence of wave structures more complex than the traveling waves presented here. 

This talk is part of the Isaac Newton Institute Seminar Series series.

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