Max-Plus Algebra for Complex Variables and Its Applications to Discrete Fourier Transformation and Partial Difference Equations

A generalization of the max-plus transformation, which is known as a method to derive cellular automata from integrable equations, is proposed for complex variables. Operation rules for this transformation is also studied for general number of variables. As an application, the max-plus transformation is applied to the discrete Fourier transformation. Ultradiscretized wave equation and nonlinear Schrödinger equation are solved by the proposed method. The solution has discrete values with inner structures in a discrete space and time.