Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

Nonlinearity [in PDEs]

The pictures below show behavior with initial conditions containing two Gaussians (and periodic boundary conditions). The diffusion and wave equations are linear, so that results are linear sums of those with single Gaussians. The sine-Gordon equation is nonlinear, but its solutions satisfy a generalized linear superposition principle. The equation from page 165 shows no such simple superposition principle. Note that even with a linear equation, fairly complicated patterns of behavior can sometimes emerge as a result of boundary conditions.

From Stephen Wolfram: A New Kind of Science [citation]