Abstract

Spatiotemporal dynamics of many natural processes, such as elasticity, heat propagation, sound waves, and fluid flows are often modeled using partial differential equations (PDEs). Certain types of PDEs have closed-form analytical solutions, some permit only numerical solutions, some require appropriate initial and boundary conditions, and others may not have stable, global, or even well-posed solutions. In this paper, we focus on one-specific type of second-order PDE — the ultrahyperbolic wave equation in multiple time dimensions. We demonstrate the wave equation solutions in complex time (kime) and show examples of the Cauchy initial value problem in space-kime. We extend the classical formulation of the dynamics of the wave equation with respect to positive real longitudinal time. The solutions to the Cauchy boundary value problem in multiple time dimensions are derived in Cartesian, polar, and spherical coordinates. These include both bounded and unbounded spatial domains. Some example solutions are shown in the main text with additional web-based dynamic illustrations of the wave equation solutions in space-kime shown in the appendix. Solving PDEs in complex time has direct connections to data science, where solving under-determined linear modeling problems or specifying the initial conditions on limited spatial dimensions may be insufficient to forecast, classify, or predict a prospective value of a parameter or a statistical model. This approach extends the notion of data observations, anchored at ordered longitudinal events, to complex time, where observables need not follow a strict positive-real structural arrangement, but instead could traverse the entire kime plane.