Published: Sept. 28, 2021

Patrick Sprenger, Department of Mathematics, North Carolina State University

Traveling wave solutions of the Kawahara equation

The Kawahara equation is an asymptotic model of weakly nonlinear wave phenomena when third and fifth order dispersion are in balance. The model equation consists of the Korteweg-de Vries equation with an additional fifth-order spatial derivative term. Traveling wave solutions of the Kawahara equation satisfy a fifth order ODE that can be integrated once, revealing the Hamiltonian structure of the resulting fourth order equation. Periodic and solitary traveling waves have generated an extensive literature related to Hamiltonian dynamics, but the fourth order traveling wave ODE allows for more general solutions including traveling waves that asymptote to distinct periodic wavetrains at infinity. Generically, these heterclinic traveling wave solutions represent a heteroclinic connections between two hyperbolic periodic orbits on a level-set of the spatial Hamiltonian and are identified by the intersection of the stable/unstable manifolds of the far-field periodic orbits.

The Hamiltonian structure of the traveling wave ODE accompanied by numerical computations of Floquet multipliers of periodic orbits reveals many such families of traveling waves. Moreover, each family bifurcates from a single elliptic, periodic traveling wave solution whose Floquet multipliers coalesce at +1. Analysis of the traveling wave Hamiltonian and numerical computations of traveling waves will be complemented by the interpretation of these solutions in terms of Whitham modulation theory. Within this framework, such heteroclinic traveling wave solutions of related models have been interpreted as discontinuous shock solutions of the Whitham modulation equations.