 Similarity solution

A similarity solution has a fixed profile after rescaling its independent variables and, when needed, its amplitude. A typical similarity ansatz isSubstitution into a partial differential equation fixes a scale balance and produces an equation for the similarity profile $F$. In one spatial dimension this is often an ordinary differential equation; in several spatial dimensions it can remain a partial differential equation unless a further symmetry, such as radiality, is imposed.

For $u_t=\Delta u$ on ${\mathbb{R}}^n$, balance of time and space derivatives gives $\beta =1/2$. Preservation of a finite nonzero spatial integral gives $\alpha =n/2$. Writing $\xi =x/\sqrt{t}$, the profile equation issolved by $F(\xi )=C{e}^{-|\xi {|}^2/4}$. The Gaussian heat kernel is therefore a spreading profile whose total mass stays fixed. A scale-invariant equation need not have a similarity solution compatible with arbitrary prescribed data; the scaling of the data and conserved quantities is part of the construction.