Stability condition and dynamic behavior for one- and two-dimensional solitary waves—Stability and interaction of two vortex lines in superfluid helium

A general stability condition and the interactions of particlelike solutions of one-dimensional sine-Gordon-type nonlinear partial differential equations are studied by numerical calculations and with the potentials which are derived from these equations. The stability condition is extended to the two-dimensional case and is applied to the study of the stability of a single vortex state and two-vortex interactions in superfluid helium near the $\lambda$ point. It was found by numerical calculations that two-vortex filaments of the same rotation repel each other, while those with opposite rotation attract each other and annihilate, contrary to classical theory.