The non-linearity of the Gross–Pitaevskii equation has its origin in the interaction between the particles: When setting the coupling constant of interaction in the Gross–Pitaevskii equation to zero (see the following section): thereby, the single-particle Schrödinger equation describing a particle inside a trapping potential is recovered. The equation has the form of the Schrödinger equation with the addition of an interaction term. The coupling constant formula_15 is proportional to the s-wave scattering length formula_11 of two interacting bosons:  where formula_18 is the reduced Planck's constant and formula_9 is the mass of the boson. The energy density is  where formula_21 is the wavefunction, or order parameter, and formula_10 is the external potential (e.g. a harmonic trap). The time-independent Gross–Pitaevskii equation, for a conserved number of particles, is  where formula_24 is the chemical potential. The chemical potential is found from the condition that the number of particles is related to the wavefunction by  From the time-independent Gross–Pitaevskii equation, we can find the structure of a Bose–Einstein condensate in various external potentials (e.g. a harmonic trap). The time-dependent Gross–Pitaevskii equation is  From the time-dependent Gross–Pitaevskii equation we can look at the dynamics of the Bose–Einstein condensate. It is used to find the collective modes of a trapped gas. Since the Gross–Pitaevskii equation is a nonlinear partial differential equation, exact solutions are hard to come by. As a result, solutions have to be approximated via myriad techniques. The simplest exact solution is the free particle solution, with formula_27,  This solution is often called the Hartree solution. Although it does satisfy the Gross–Pitaevskii equation, it leaves a gap in the energy spectrum due to the interaction:  According to the Hugenholtz–Pines theorem,  an interacting bose gas does not exhibit an energy gap (in the case of repulsive interactions). A one-dimensional soliton can form in a Bose–Einstein condensate, and depending upon whether the interaction is attractive or repulsive, there is either a bright or dark soliton. Both solitons are local disturbances in a condensate with a uniform background density.