Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems.
Electron. J. Diff. Eqns., Conference 21 (2014), pp. 61-76.

Symmetry analysis and numerical solutions for semilinear elliptic systems

C. Tyler Diggans, John M. Neuberger, James W. Swift

Abstract:
We study a two-parameter family of so-called Hamiltonian systems defined on a region $\Omega$ in $\mathbb{R}^d$ with the bifurcation parameters $\lambda$ and $\mu$ of the form:
$$\displaylines{ \Delta u + \frac{\partial}{\partial v}H_{\lambda,\mu}(u,v)=0 \quad \text{in } \Omega,\cr \Delta v + \frac{\partial}{\partial u}H_{\lambda,\mu}(u,v)=0, \quad \text{in } \Omega }$$
taking $H_{\lambda,\mu}$ 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.

Published February 10, 2014.
Math Subject Classifications: 35J15, 65N30.
Key Words: Nonlinear elliptic PDE; elliptic systems; Newton's method; GNGA; bifurcation.

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C. Tyler Diggans
Department of Mathematics
Northern Arizona University, 86005
Flagstaff, AZ 86011, USA
email: Tyler.Diggans@nau.edu
John M. Neuberger
Department of Mathematics
Northern Arizona University, 86005
Flagstaff, AZ 86011, USA
email: John.Neuberger@nau.edu
  Jim W. Swift
Department of Mathematics
Northern Arizona University, 86005
Flagstaff, Arizona, USA
email: Jim.Swift@nau.edu

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