Abstract
We give a method for computing nontrivial solutions to the nonlinear partial differential equation ∇ 2 Ψ + λ 2sinh Ψ = 0, with Ψ = 0 on a square boundary. The method consists of a Newton-Raphson iteration, in which successive corrections to Ψ must satisfy a linearized partial differential equation. We give a direct solution algorithm for the linearized equation, which is suitable for small meshes. Using this method, we establish the nonuniqueness of solutions by finding six solutions for the same value of λ. Calculation of these solutions required from 4 to about 40 iterations each, depending upon the accuracy of the initial approximation. These solutions are the lowest-mode members of three classes of solutions possessing (1) rectangular, (2) quasicylindrical, and (3) diagonal symmetry.
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