On non-existence of bifurcations in one-dimensional Bratu equation

TL;DR

This paper analyzes the one-dimensional Bratu equation, highlighting that spurious bifurcations appear in discretized schemes (FE and FD) even when symmetry is enforced, while providing a rigorous analytical result that the continuous problem has no bifurcations along the nontrivial solution branch. It parameterizes nontrivial solutions to derive a precise condition for critical points and uses bifurcation theory to show that only a limit point exists on the primary branch. The findings stress the importance of verifying numerical bifurcations in nonlinear boundary value problems and explain the origin of antisymmetric modes observed in discretizations. Overall, the work distinguishes numerical artifacts from true bifurcation structure in a classical nonlinear PDE setting.

Abstract

In this paper, we revisit the classical problem of Bratu differential equation in one-dimension. While it is known that the finite difference discretized form of continuous Bratu equation gives rise to spurious bifurcations, we show that spurious bifurcation points exist even when the finite element approach is employed. We then present an analytical proof demonstrating that there are no bifurcations when the continuous Bratu equation is considered.

Figures (1)

• Figure 1: Bifurcation diagram obtained using finite element (a) and finite difference (b) discretizations. The bifurcation points are marked with a red dot on the solution path. (c) Characteristic sawtooth form of the eigenvector $w(x)$ at the bifurcation point for $N = 101$.