Cole–Hopf transformation

The Cole–Hopf transformation is a change of variables that allows to transform a special kind of parabolic partial differential equations (PDEs) with a quadratic nonlinearity into a linear heat equation. In particular, it provides an explicit formula for fairly general solutions of the PDE in terms of the initial datum and the heat kernel.

Consider the following PDE:$$u\_\{t\}\; -\; a\backslash Delta\; u\; +\; b\backslash |\backslash nabla\; u\backslash |^\{2\}\; =\; 0,\; \backslash quad\; u(0,x)\; =\; g(x)$$

where $x\backslash in\; \backslash mathbb\{R\}^\{n\}$, $a,b$ are constants, $\backslash Delta$ is the Laplace operator, $\backslash nabla$ is the gradient, and $\backslash |\backslash cdot\backslash |$ is the Euclidean norm in $\backslash mathbb\{R\}^\{n\}$. By assuming that $w\; =\; \backslash phi(u)$, where $\backslash phi(\backslash cdot)$ is an unknown smooth function, we may calculate:$$w\_\{t\}\; =\; \backslash phi\text{'}(u)u\_\{t\},\; \backslash quad\; \backslash Delta\; w\; =\; \backslash phi\text{'}(u)\backslash Delta\; u\; +\; \backslash phi\text{'}\text{'}(u)\backslash |\backslash nabla\; u\backslash |^\{2\}$$

Which implies that:$$\backslash begin\{aligned\}$$

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

{{Equation box 1|cellpadding|border|indent=:|equation=$w(u)\; =\; e^\{-bu/a\}$|border colour=#0073CF|background colour=#F5FFFA}}This is the Cole-Hopf transformation.{{Cite book |last=Evans |first=Lawrence C. | authorlink=Lawrence C. Evans|title=Partial Differential Equations |publisher=American Mathematical Society |year=2010 |isbn= |edition=2nd |series=Graduate Studies in Mathematics |volume=19 |pages=206–207}} With the transformation, the following initial-value problem can now be solved:$$w\_\{t\}\; -\; a\backslash Delta\; w\; =\; 0,\; \backslash quad\; w(0,x)\; =\; e^\{-bg(x)/a\}$$

The unique, bounded solution of this system is:$$w(t,x)\; =\; \{1\backslash over\{(4\backslash pi\; at)^\{n/2\}\}\}\; \backslash int\_\{\backslash mathbb\{R\}^\{n\}\}\; e^\{-\backslash |x-y\backslash |^\{2\}/4at\; -\; bg(y)/a\}dy$$

Since the Cole–Hopf transformation implies that $u\; =\; -(a/b)\backslash log\; w$, the solution of the original nonlinear PDE is:$$u(t,x)\; =\; -\{a\backslash over\{b\}\}\backslash log\; \backslash left[\; \{1\backslash over\{(4\backslash pi\; at)^\{n/2\}\}\}\; \backslash int\_\{\backslash mathbb\{R\}^\{n\}\}\; e^\{-\backslash |x-y\backslash |^\{2\}/4at\; -\; bg(y)/a\}dy\; \backslash right]$$

The complex form of the Cole-Hopf transformation can be used to transform the Schrödinger equation to the Madelung equation.{{Cite journal |last=Madelung |first=E. |date=1927 |title=Quantentheorie in hydrodynamischer Form |journal=Die Naturwissenschaften (In German) |volume=40 |issue=3–4 |pages=322–326 |doi=10.1007/BF01504657 |issn=1434-6001}}