  Coupled oscillators can serve as a testbed for larger questions of pattern
formation across many areas of science and engineering. Much effort has been
dedicated to the Kuramoto model and phase oscillators, but less has focused on
oscillators with variable amplitudes. Here we examine the simplest such
oscillators -- Stuart-Landau oscillators -- and attempt to elucidate some
puzzling dynamics observed in simulation by us and others. We demonstrate the
existence and stability of a previously unreported state which we call a
``phase chimera state.'' Remarkably, in this state, the amplitudes of all
oscillators are identical, but one subset of oscillators phase-locks while
another subset remains incoherent in phase. We also show that this state can
take the form of a ``multitailed phase chimera state'' where a single
phase-synchronous cluster of oscillators coexists with multiple groups of
phase-incoherent oscillators.
