Bright traveling breathers in media with long-range, nonconvex dispersion

TL;DR

This paper investigates bright traveling breathers in nonlocal dispersive media, revealing their existence, properties, and how they act as defects on periodic backgrounds, with implications for oceanic and optical systems.

Contribution

It introduces a numerical method to find bright traveling breather solutions in nonlocal models with nonconvex dispersion, highlighting their topological and resonance features.

Findings

Traveling breathers exist on periodic backgrounds with nonzero oscillations.

Solutions exhibit a topological phase jump, acting as defects.

Breathers are approximated by bright solitons at small amplitudes.

Abstract

The existence and properties of envelope solitary waves on a periodic, traveling wave background, called traveling breathers, are investigated numerically in representative nonlocal dispersive media. Using a fixed point computational scheme, a space-time boundary value problem for bright traveling breather solutions is solved for the weakly nonlinear Benjamin-Bona-Mahony equation, a nonlocal, regularized shallow water wave model, and the strongly nonlinear conduit equation, a nonlocal model of viscous core-annular flows. Curves of unit-mean traveling breather solutions within a three-dimensional parameter space are obtained. Resonance due to nonconvex, rational linear dispersion leads to a nonzero oscillatory background upon which traveling breathers propagate. These solutions exhibit a topological phase jump, so act as defects within the periodic background. For small amplitudes, traveling breathers are well-approximated by bright soliton solutions of the nonlinear Schr\"odinger equation with a negligibly small periodic background. These solutions are numerically continued into the large amplitude regime as elevation defects on cnoidal or cnoidal-like periodic traveling wave backgrounds. This study of bright traveling breathers provides insight into systems with nonconvex, nonlocal dispersion that occur in a variety of media such as internal oceanic waves subject to rotation and short, intense optical pulses.