Cusp bifurcation in the eigenvalue spectrum of PT-symmetric Bose-Einstein condensates

A Bose-Einstein condensate in a double-well potential features stationary solutions even for attractive contact interactions as long as the particle number and therefore the interaction strength does not exceed a certain limit. Introducing balanced gain and loss into such a system drastically changes the bifurcation scenario at which these states are created. Instead of two tangent bifurcations at which the symmetric and antisymmetric states emerge, one tangent bifurcation between two formerly independent branches arises [Phys. Rev. A 89, 023601 (2014)]. We study this transition in detail using a bicomplex formulation of the time-dependent variational principle and find that, in fact, there are three tangent bifurcations for very small gain-loss contributions which coalesce in a cusp bifurcation.