I am interesting in solving the following nonlinear, time-dependent pde in 2 spatial dimensions (complex Gross-Pitaevskii eq):$$i \frac{\partial \psi}{\partial t} = \left[ -\nabla^2 + (1-i \sigma)(|\psi|^2-1) \right] \psi$$The goal is to find steady state solutions for the function $\psi(x,y,t)$, for different parameters $\sigma$. Here $\sigma$ is a positive real-valued parameter. The boundary conditions can initially be of the Dirichlet type, setting the function $\psi$ to zero on the contour of a square, but later on I plan to implement some absorbing BC (perhaps using the perfectly matched layer method?). I am also considering solving the equation in a different geometry later on, on a disk using polar coordinates.For the time-dependence, I plan to use the Gryphon module of Erik Skare (https://launchpad.net/gryphonproject), which is basically a Runge-Kutta solver and for the spatial part Fenics.So the question is, do you think this is feasible to do with Fenics, and if so, how would one proceed?
time-dependent nonlinear pde in fenics
pde;fenics
I would try pushing this formV = VectorFucntioSpace(mesh, 'CG', 1, dim=2)u = Function(V)v = TestFunction(V)sigma = Constant(0.1)F = inner(grad(u), grad(v))*dx \  + (inner(u, u)-1.0)*((u[0]+sigma*u[1])*v[0] + (u[1]-sigma*u[0])*v[1])*dxinto nonlinear-poisson demo. But note that this would yield trivial solution if you start from zero field with zero Dirichlet.