Gross-Pitaevski Equation

In this example we use the Newton’s method to solve the nonlinear complex-valued time-dependent Gross-Pitaevski equation. This equation describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model. For time-discretization one can use either the first-order implicit Euler method or the second-order Crank-Nicolson method.

Problem description

The computational domain is the square (-1,1)^2 and boundary conditions are zero Dirichlet. The equation has the form

i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \Delta \psi + g \psi |\psi|^2 + \frac{m}{2} \omega^2 (x^2 + y^2) \psi

where \psi(x,y) is the unknown solution (wave function), i the complex unit, \hbar the Planck constant, m the mass of the boson, g the coupling constant (proportional to the scattering length of two interacting bosons) and \omega the frequency.

Sample results

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