  We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell,
and subjected to a tilted nonuniform magnetic field. It is shown that the
interface between the ferrofluid and an inviscid outer fluid (air) supports
traveling waves, governed by a novel modified Kuramoto--Sivashinsky-type
equation derived under the long-wave approximation. The balance between energy
production and dissipation in this long-wave equations allows for the existence
of dissipative solitons. These permanent traveling waves' propagation velocity
and profile shape are shown to be tunable via the external magnetic field. A
multiple-scale analysis is performed to obtain the correction to the linear
prediction of the propagation velocity, and to reveal how the nonlinearity
arrests the linear instability. The traveling periodic interfacial waves
discovered are identified as fixed points in an energy phase plane. It is shown
that transitions between states (wave profiles) occur. These transitions are
explained via the spectral stability of the traveling waves. Interestingly,
multiperiodic waves, which are a non-integrable analog of the double cnoidal
wave, are also found to propagate under the model long-wave equation. These
multiperiodic solutions are investigated numerically, and they are found to be
long-lived transients, but ultimately abruptly transition to one of the stable
periodic states identified.
