SYMMETRY ANALYSIS AND NUMERICAL SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS

We study a two-parameter family of so-called Hamiltonian systems defined on a region Omega in R-d with the bifurcation parameters lambda and mu of the form: Delta u + partial derivative/partial derivative v H-lambda,H-mu (u, v) = 0 in Omega, Delta v + partial derivative/partial derivative u H-lambda,H-mu (u, v) = 0, in Omega taking H-lambda,H-mu to be a function of two variables satisfying certain conditions. We use numerical methods adapted from Automated Bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs (Inter. J. Bif. Chaos, 2009) to approximate solution pairs. After providing a symmetry analysis of the solution space of pairs of functions defined on the unit square, we numerically approximate bifurcation surfaces over the two dimensional parameter space. A cusp catastrophe is found on the diagonal in the parameter space where lambda = mu and is explained in terms of symmetry breaking bifurcation. Finally, we suggest a more theoretical direction for our future work on this topic.