Twenty-five years of dissipative solitons

In 1995, C. I. Christov and M. G. Velarde introduced the concept of a dissipative soliton in long-wave thin-film equation [Physica D 86, 323–347]. 25 years since, subject has blossomed to include many related phenomena. The focus this short note is survey conceptual influence “production-dissipation (input-output) energy balance” that they identified. Our recent results on nonlinear periodic waves as solitons (in model for ferrofluid interface parallel-flow rectangular geometry an inhomogeneous magnetic field) have shown classical also applies nonlocalized (specifically, spatially periodic) coherent structures. Thus, we revisit so-called KdV-KSV studied by demonstrate it possesses solutions. These structures arise when linearly unstable flat film state evolves sufficiently large amplitude. linear instability then arrested nonlinearity saturates, leading permanent traveling waves. Although two equations considered feature same prototypical mechanism, along with similar dispersion, their nonlinearities are fundamentally different. terms set shape eventual dynamics Intriguingly, nonintegrable discussed exhibit multiperiodic wave solutions, akin polycnoidal J. P. Boyd context completely integrable KdV equation.