Dynamics of solitary waves on a ferrofluid jet: the Hamiltonian framework

The stability and dynamics of solitary waves propagating along the surface of an inviscid ferrofluid jet in the absence of gravity are investigated analytically and numerically. For the axisymmetric geometry, the problem is shown to be a conservative system with total energy as the Hamiltonian; however, one of the canonical variables differs from those in the classic water-wave problem in the Cartesian coordinate system. The Dirichlet–Neumann operator appearing in the kinetic energy is then expanded as a Taylor series, described in homogeneous powers of the surface displacement. Based on the further analysis of the Dirichlet–Neumann operator, a systematic procedure is proposed to derive reduced model equations of multiple scales in various asymptotic limits from the full Euler equations in the Hamiltonian/Lagrangian framework. Particularly, a fully dispersive model arising from retaining terms valid up to the quartic order in the series expansion of the kinetic energy, which results in quadratic and cubic algebraic nonlinearities in Hamilton's equations and henceforth is abbreviated as the cubic full-dispersion model, is proposed. By comparing bifurcation curves and wave profiles of various types of axisymmetric solitary waves among different model equations, the cubic full-dispersion model is found to agree well with the full Euler equations, even for waves of considerably large amplitudes. The stability properties of axisymmetric solitary waves subjected to longitudinal disturbances are verified with the newly proposed model. Our analytical results, consistent with Saffman's theory, indicate that in the axisymmetric cylindrical system, the stability exchange subjected to superharmonic perturbations also occurs at the stationary point of the speed-energy bifurcation curve. A series of numerical experiments for the stability and dynamics of solitary waves are performed via the numerical time integration of the model equation, and collision interactions between stable solitary waves show non-elastic features.