In this work we solve the Dirac equation by constructing the exact bound state solutions for a mixing of generalized vector and scalar Hartmann potentials. This is done provided the vector and scalar potentials hold some relation. Namely, one must be equals to or minus the other. Finally the case of some quasi-exactly solvable potentials are briefly commented.

Dark solitons are some of the most interesting nonlinear excitations and considered to be one-dimensional topological analogs vortices. However, in contrast their two-dimensional vortex counterparts, characteristics a dark soliton far from fully understood because charge defined according phase jump cannot reflect its essential property. Here, similar complex extension used exploration partitio...

We derive a Hodge decomposition of discrete vector fields on polyhedral surfaces, and apply it to the identification of vector field singularities. This novel approach allows us to easily detect and analyze singularities as critical points of corresponding potentials. Our method uses a global variational approach to independently compute two potentials whose gradient respectively co-gradient ar...