We consider the problem of numerically computing solutions of evolutionary nonlinear partial differential equations (PDEs) with a finite-dimensional group of symmetries. Specifically, we look for solutions that are fixed by elements of the equations' symmetry group. The latter class includes time-periodic solutions. We work with the complex Ginzburg-Landau equation (CGLE) in one space dimension, which has a 3-parameter group of symmetries generated by space-time translations and a rotation of the (complex) amplitude. The spectral-Galerkin method used to discretize the PDE will be described, along with the approach for solving the resulting system of nonlinear algebraic equations which allowed us to identify multiple new solutions in a chaotic region of the CGLE.
Due to the relatively small number of unknowns considered (2,000 - 3,000 after discretization), it was possible to use a direct method for linear systems as part of the process for solving the nonlinear system. However, for problems with a large number of unknowns, iterative methods for linear systems are required. We will conclude our talk with a discussion on the use of such methods for solving these types of problems.
