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1 Problem setup

One approach to understand wave dynamics in spacekime is to plot the basic separable solution. Recall the 5D wave equation. \[ \Delta_x u = \Delta_t u, \text{ where } x=(x_1,x_2,x_3)\in\mathbb{R}^3, t=(t_1,t_2)\in\mathbb{R}^2\] The equation is essentially a ultrahyperbolic equation, and in general, if we do not impose a proper condition, the equation does not permit stable and unique and global solutions. One possible approach to resolve these instabilities of the potential functions is to impose Periodic Boundary conditions (PBC) and consider the corresponding base function solutions. In general, the basis takes the form \[u(x,t)=e^{2\pi i(x_1\xi_1+x_2\xi_2+x_3\xi_3+t_1\eta_1+t_2\eta_2) }, \xi_1,\xi_2,\xi_3,\eta_1,\eta_2\in \mathbb{Z}\] For the periodic boundary conditions to hold, we require that “multiples” of the spatial and temporal coordinates are integers. In the simplest scenario, we consider a “degenerated” hyperbolic equation to visualize the dynamics of wave equation under Periodic boundary conditions on the spatial dimension, a simpler case in 1 spatial dimension and 1 time dimension is: \[ \Delta_x u = \Delta_t u, x\in \mathbb{R}, t\in \mathbb{R}, u(-1,t)=u(1,t), u_x(-1,t)=u_x(1,t), u(x,0)=Cos(\pi x), u_t(x,0)=-\pi Sin(\pi x)\]