Assume a lower-dimensional solitonic structure embedded in a higher
dimensional space, e.g., a 1D dark soliton embedded in 2D space, a ring dark
soliton in 2D space, a spherical shell soliton in 3D space etc. By extending
the Landau dynamics approach [Phys. Rev. Lett. {\bf 93}, 240403 (2004)], we
show that it is possible to capture the transverse dynamical modes (the "Kelvin
modes") of the undulation of this "soliton filament" within the higher
dimensional space. These are the transverse stability/instability modes and are
the ones potentially responsible for the breakup of the soliton into structures
such as vortices, vortex rings etc. We present the theory and case examples in
2D and 3D, corroborating the results by numerical stability and dynamical
computations.Comment: 5 pages, 3 figure
