The nonlinear heat equation u_t = u_xx + u^2 with smooth periodic initial data blows up in finite time. Continuing the solution to complex values of the spatial variable turns that event into the motion of singularities: for this quadratic nonlinearity they are movable logarithmic branch points, born off the real axis and drawn toward it, and blow-up occurs exactly when a singularity reaches the real line. Learners cover how such singularities are located and tracked numerically -- Fourier spectral discretisation combined with quadratic Fourier-Pade analytic continuation, and a pole field solver that integrates the governing ODE onto neighbouring Riemann sheets -- and how matched asymptotic expansions on several separated time scales describe the approach to blow-up: an initial linear diffusive phase, the onset of nonlinearity, a self-similar regime whose corrections are organised by Hermite polynomial eigenmodes, and the final collapse of the singularity onto the real axis. The pay-off is a single geometric picture in which blow-up time, rate and location are read off the trajectory of a complex singularity rather than inferred from the real-line solution alone.
The nonlinear heat equation u_t = u_xx + u^2, taken with 2pi-periodic initial data u(x,0) = alpha*cos(x), develops finite-time blow up. Continued into the complex spatial plane z = x + iy, the soluti…