The Kawahara Equation: Traveling Wave Solutions Joining Periodic Waves

TL;DR

This paper investigates traveling wave solutions of the Kawahara equation, deriving conditions for connecting periodic waves and constructing heteroclinic orbits that represent solitary waves bridging different periodic states.

Contribution

It introduces a framework for analyzing heteroclinic traveling waves in the Kawahara equation using bifurcation theory and manifold intersections, revealing new solution structures.

Findings

Derived jump conditions for periodic wave pairs

Constructed multiple solution branches via bifurcation analysis

Identified heteroclinic orbits connecting periodic solutions

Abstract

The Kawahara equation is a weakly nonlinear long-wave model of dispersive waves that emerges when leading order dispersive effects are in balance with the next order correction. Traveling wave solutions of the Kawahara equation satisfy a fourth-order ordinary differential equation in which the traveling wave speed is a parameter. The fourth order equation has Hamiltonian structure and admits a two-parameter family of single-phase periodic solutions with varying speed and Hamiltonian. A set of jump conditions is derived for pairs of periodic solutions with equal speed and Hamiltonian. These are necessary conditions for the existence of traveling waves that asymptote to the periodic orbits at $\pm \infty$. Bifurcation theory and parameter continuation are used to construct multiple solution branches of the jump conditions. For each pair of compatible periodic solutions, the heteroclinic orbit representing the traveling wave is constructed from the intersection of stable and unstable manifolds of the periodic orbits. 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.