a self-similar solution
Watch a drop of ink spread in still water. Photograph it after one second, after four seconds, after nine. If you zoom out on the later photos by just the right amount, they look identical to the first — same shape, just bigger. The spreading does not invent new shapes; it rescales one fixed shape over and over. A self-similar solution is a solution that does exactly this: it reproduces itself under a coordinated stretch of space and time. It is the same profile getting wider and shorter.
Precisely: a self-similar solution has the form u(x, t) = t^(-alpha) F(x / t^(beta)) for some profile function F and exponents alpha, beta. The single combined variable eta = x / t^(beta) is called the similarity variable, and F(eta) is the universal profile. The payoff is huge: substituting this ansatz into a PDE in two variables (x and t) collapses it into an ORDINARY differential equation for F in the one variable eta — the dimensions of the problem drop, and an ODE is far easier to solve or analyze. For the heat equation u_t = k u_xx, the right exponents are beta = 1/2 and alpha = 1/2, and the resulting profile F is the Gaussian: the heat kernel itself is the fundamental self-similar solution of diffusion. The 1/2 says diffusion spreads like the square root of time, the signature of a random walk.
Self-similar solutions matter far beyond being easy to compute: they describe the long-time behaviour of broad classes of problems. Very often a solution from generic, messy initial data forgets its details and converges, after rescaling, to a self-similar profile — the self-similar solution is an attractor for the dynamics. This is why diffusion from almost any localized initial heat eventually looks Gaussian. They also organize the study of blow-up (where a nonlinear solution becomes infinite in finite time, often in a self-similar way) and of fronts and intermediate asymptotics. Honesty note: a self-similar form only exists when the equation and data have a scaling symmetry; not every problem admits one.
Seek u(x, t) = t^(-1/2) F(x / sqrt(t)) for the heat equation. Plugging in turns u_t = k u_xx into an ODE for F whose solution is a Gaussian — the heat kernel. Every localized blob of heat eventually relaxes to this same self-similar Gaussian shape, just rescaled.
One scaling variable turns a PDE into an ODE — and names the long-time attractor.
A self-similar solution exists only when the equation and the data share a scaling symmetry; without it, the ansatz simply will not reduce the PDE to an ODE. When it does exist, do not assume every solution converges to it — that the self-similar profile is an attractor is a separate fact that must be proved.