On the interaction between two nonpropagating hydrodynamic solitons

Abstract

The evolution equation governing the positions of two interacting nonpropagating solitons is found by using the variational principle approach. Furthermore, the potential which characterizes the interaction between solitons is derived, from which some important results are obtained: (1) Two solitons with the same polarity always attract each other, while those with opposite polarity repel. (2) The repulsion is much weaker than the attraction and both attraction and repulsion are of short distance. (3) The period of mutual oscillation caused by interaction between two solitons with the same polarity increases as the drive amplitude rises; especially, the period can spread from zero to infinity as the initial distance between solitons increases. All the above results are well coincident with experiments.

Non-integrable dynamics of matter-wave solitons in a density-dependent gauge theory

“…In the first instance, we approximate the two-soliton state as a linear superposition of two individual solitons, with the interaction treated as the spatial overlap of the soliton envelopes. This technique has been previously applied to interaction problems in the NLS [53,54] and Gross-Pitaevskii equations [55,56], in addition to several others [57][58][59][60][61]. An advantage of this method is the ability to derive a set of variational equations which describe the motion of the solitons.…”

“…In this appendix we show how to evaluate the interaction integrals which appear in the variational analysis, using the method of residues. These calculations can be found in the standard literature [54,57] , but we recapitulate details here, to provide a basis for evaluating more complicated integrals.…”