Compact breathers in a quasi-linear Klein–Gordon equation

We study the quasi-linear complex Klein–Gordon equation −Ztt+∇(|∇Z|2∇Z)=P′(|Z|)Z|Z| and present two classes of strictly localized compact stationary breathers. In the first class breathers vibrate at an anharmonic rate but the site potential has to be quartic. In the second class a more general, Q-ball type, site potentials are admitted but vibrations are harmonic. Notably, unlike the Q-balls supporting models, if the chosen potential has a top then multi-nodal modes cannot accumulate there: only a finite number of multi-nodal modes is possible, each constrained by its own spectrum of harmonic vibrations.