The "self-focusing" nonlinear Schrödinger equation
, and so on, also known as the Gross–Pitaevskii equation, appears in wave propagation in nonlinear media such as signal transmission in optical fibers, Bose–Einstein condensation, and surface waves over sufficiently deep water. In addition, this model plays a major role in the theory of rogue waves. After the Korteweg–de Vries equation, this was the second nonlinear partial differential equation whose initial value problem was solved via the inverse scattering transform method. It has two types of solutions, one associated with a group of waves and the other in which the velocity depends on the amplitude. The first is a soliton whose envelope does not change in form over time. The second family of solutions are the so-called "breathers", which are rational solutions of the nonlinear Schrödinger equation.
In this Demonstration a spatially localized algebraic breather solution with only one oscillation is studied. The squared modulus
of the wavefunction is interpreted as the particle density, where
is a complex function and
is its complex conjugate. The causal interpretation of quantum theory developed by Louis de Broglie and David Bohm introduced trajectories that are guided by a real phase function determined by the wavefunction. In the eikonal representation of the wave,
, the gradient of the phase
is the particle velocity
. In this interpretation, the origin of the motion of the particle is the potential
, given by
, plus an additional term
. The latter may be interpreted alternatively as a kind of fluid pressure or quantum Bohm potential. In the asymptotic case, the velocity of the particle becomes
. In the graphic on the left, you can see the position of the particles, the squared wavefunction (blue), the quantum potential (red), and the velocity (green). In the center and on the right, the graphics show the squared wavefunction plus trajectories and the quantum potential plus trajectories.