Dynamics of nonlinear Schrödinger breathers in a potential trap

We consider the evolution of the 2-soliton (breather) of the nonlinear Schrödinger equation on a semi-infinite line with the zero boundary condition and a linear potential, which corresponds to the gravity field in the presence of a hard floor. This setting can be implemented in atomic Bose-Einstein condensates, and in a nonlinear planar waveguide in optics. In the absence of the gravity, repulsion of the breather from the floor leads to its splitting into constituent fundamental solitons, if the initial distance from the floor is smaller than a critical value; otherwise, the moving breather persists. In the presence of gravity, the breather always splits into a pair of "co-hopping" fundamental solitons, which may be frequency locked in the form of a quasi-breather, or unlocked, forming an incoherent pseudo-breather. Some essential results are obtained in an analytical form, in addition to the systematic numerical investigation.