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  "body": "\n## Summary\n\nThe Cole–Hopf transformation turns the viscous [[Burgers' equation|NEGHXN8RE1]] into the linear [[heat equation|VP2HN86VAG]].\nIt sets $u = -2\\nu\\, \\partial_x \\log \\varphi$, where $\\varphi$ is a positive solution of the heat equation.\nIt gives an explicit formula for the solution of the initial value problem for Burgers' equation.\nLetting the viscosity tend to zero in that formula recovers the entropy solution of the inviscid equation.\nSee the [[definition|PPM7V7WWHQ]].\n\n## Explanations\n\n### L1 human\n\nBurgers' equation is nonlinear, so solutions cannot simply be added together.\nThe Cole–Hopf trick is a change of unknown that removes the nonlinearity.\nAfter the change, the equation becomes the heat equation, which is solved by a known integral formula.\nChanging back gives the solution of Burgers' equation with positive viscosity for given initial values.\nFor example, $\\varphi(x, t) = x$ solves the heat equation, and it gives the solution $u = -2\\nu / x$ of Burgers' equation for $x \u003e 0$.\n",
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