Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: A Taylor-Chebyshev series approach

Abstract

In this paper, we introduce a rigorous computational approach to prove existence of rotation invariant patterns for a nonlinear Laplace-Beltrami equation posed on the 2-sphere. After changing to spherical coordinates, the problem becomes a singular second order boundary value problem (BVP) on the interval \begin{document}$ (0,\frac{\pi}{2}] $\end{document} with a removable singularity at zero. The singularity is removed by solving the equation with Taylor series on \begin{document}$ (0,\delta] $\end{document} (with \begin{document}$ \delta $\end{document} small) while a Chebyshev series expansion is used to solve the problem on \begin{document}$ [\delta,\frac{\pi}{2}] $\end{document}. The two setups are incorporated in a larger zero-finding problem of the form \begin{document}$ F(a) = 0 $\end{document} with \begin{document}$ a $\end{document} containing the coefficients of the Taylor and Chebyshev series. The problem \begin{document}$ F = 0 $\end{document} is solved rigorously using a Newton-Kantorovich argument.