Cole–Hopf transformation

The Cole–Hopf transformation is a method of solving parabolic partial differential equations (PDEs) with a quadratic nonlinearity of the form:

$${\displaystyle u_{t}-a\Delta u+b\|\nabla u\|^{2}=0,\quad u(0,x)=g(x)}$$

where ${\displaystyle x\in \mathbb {R} ^{n}}$, ${\displaystyle a,b}$ are constants, ${\displaystyle \Delta }$ is the Laplace operator, ${\displaystyle \nabla }$ is the gradient, and ${\displaystyle \|\cdot \|^{2}}$ is the ${\displaystyle \ell ^{2}}$-norm. By assuming that ${\displaystyle w=\phi (u)}$, where ${\displaystyle \phi (\cdot )}$ is an unknown smooth function, we may calculate:

$${\displaystyle w_{t}=\phi '(u)u_{t},\quad \Delta w=\phi '(u)\Delta u+\phi ''(u)\|\nabla u\|^{2}}$$

Which implies that:

$${\displaystyle {\begin{aligned}w_{t}=\phi '(u)u_{t}&=\phi '(u)\left(a\Delta u-b\|\nabla u\|^{2}\right)\\&=a\Delta w-(a\phi ''+b\phi ')\|\nabla u\|^{2}\\&=a\Delta w\end{aligned}}}$$

if we constrain ${\displaystyle \phi }$ to satisfy ${\displaystyle a\phi ''+b\phi '=0}$. Then we may transform the original nonlinear PDE into the canonical heat equation by using the transformation:

This is the Cole-Hopf transformation.[1] With the transformation, the following initial-value problem can now be solved:

$${\displaystyle w_{t}-a\Delta w=0,\quad w(0,x)=e^{-bg(x)/a}}$$

The unique, bounded solution of this system is:

$${\displaystyle w(t,x)={1 \over {(4\pi at)^{n/2}}}\int _{\mathbb {R} ^{n}}e^{-\|x-y\|^{2}/4at-bg(y)/a}dy}$$

Since the Cole–Hopf transformation implies that ${\displaystyle u=-(a/b)\log w}$, the solution of the original nonlinear PDE is:

$${\displaystyle u(t,x)=-{a \over {b}}\log \left[{1 \over {(4\pi at)^{n/2}}}\int _{\mathbb {R} ^{n}}e^{-\|x-y\|^{2}/4at-bg(y)/a}dy\right]}$$