Exact solvability of the Gross-Pitaevskii equation for general potentials with bound states
DOI:
https://doi.org/10.31349/RevMexFis.72.040401Keywords:
Bound states, Solitons, Gross-Pitaevskii equation, IntegrabilityAbstract
In this paper we present the analytic solution of the bound state problem for the Gross-Pitaevskii (GP) equation in 1D and its properties, in the presence of external potentials such as finite square wells and attractive Dirac deltas, as well as stable solitons for repulsive defects. We show that the GP equation can be mapped to a first-order non-autonomous dynamical system, whose solutions can sometimes be written in terms of known functions. The formal solutions of this non-conservative system can be written with the help of Glauber-Trotter formulas or a series of ordered exponentials in the coordinate x. With this we illustrate how to solve any nonlinear problem based on a construction due to Mello and Kumar for the linear case (layered potentials). For the benefit of the reader, we comment on the difference between the integrability of a quantum system and the solvability of the wave equation.
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