a self-similar diffusion solution
Photograph an ink blot spreading in water at two different times. If you zoom in on the later, wider blot by just the right amount, it looks identical to the earlier one. Diffusion has a hidden symmetry: its spreading profiles at different times are scaled copies of a single shape. A self-similar solution is one that exploits this symmetry to collapse the PDE down to an ODE.
The heat equation u_t = k u_xx has a scaling symmetry: if u(x,t) solves it, so does u(a x, a^2 t) for any a > 0 — space scales like the square root of time, the x ~ sqrt(k t) law again. This suggests looking for solutions that depend on x and t only through the single similarity variable eta = x / sqrt(4 k t). The guess u(x,t) = t^(-1/2) F(eta) (the power chosen so total heat is conserved) turns the PDE into an ordinary differential equation for F(eta): F'' + 2 eta F' + 2 F = 0. Solving it, with the condition that the total heat is one, gives F(eta) proportional to e^(-eta^2) — and putting it back together you recover exactly the heat kernel, the spreading Gaussian. So the fundamental solution is itself the self-similar solution from a point source. The method, called dimensional analysis or the similarity method, works because the bare heat equation has no built-in length or time scale, only the diffusivity k.
Self-similar solutions are powerful far beyond the heat equation. Whenever a problem has a scaling symmetry and no fixed scale, you can often reduce dimensions this way — a PDE in two variables becomes an ODE in one — and the resulting profile is the universal long-time shape that a wide class of initial data converges to. For diffusion, the point-source Gaussian is exactly that universal attractor: start with any localized lump of heat and after enough time it looks like a spreading Gaussian, having forgotten the details of its initial shape. Caveat: self-similarity requires the scaling symmetry to be unbroken — a finite interval, a fixed length in the boundary conditions, or a source at a particular scale breaks it, and then no exact similarity solution exists.
Look for u = t^(-1/2) F(x/sqrt(4 k t)); the heat equation collapses to F'' + 2 eta F' + 2 F = 0, whose unit-mass solution is the Gaussian e^(-eta^2)/sqrt(4 pi k t) — the heat kernel itself.
Two variables (x,t) become one (eta) — a PDE becomes an ODE.
Self-similarity needs the scaling symmetry to survive. A finite domain, a fixed boundary length, or a scale-carrying source breaks it, and then the exact similarity solution no longer applies (though it may still describe intermediate-time behaviour).