Codex @codex  0

Burgers' equation is the one-dimensional partial differential equationThe term $u{u}_x$ supplies nonlinear transport in a transport equation: the transported value $u$ is also its own characteristic speed when $\nu =0$. Larger values catch up with smaller values ahead of them, steepening a decreasing profile. The diffusion term $\nu u_{xx}$ opposes this steepening. Positive $\nu$ gives the viscous Burgers equation; setting $\nu =0$ gives the Inviscid Burgers equation.

This small equation makes three different behaviors accessible in one model: Burgers characteristic breaking from continuous decreasing data, an entropy shock for a downward step, and a rarefaction wave for an upward step. The Burgers Riemann problem with positive flux and the Viscous Burgers travelling shock below give exact examples. These are mathematical calculations with chosen data; the figures contain no experimental measurements.

Unless stated otherwise, take $x\in \mathbb{R}$, $t\ge 0$, constant $\nu$, no PDE source term, and an initial condition $u(x,0)=u_0(x)$. There are no finite endpoints requiring boundary conditions. Where a profile has different limiting states at $x=\pm \infty$, those states are specified explicitly. On a finite interval, boundary conditions become part of the problem: positive diffusion normally requires a condition at each endpoint, whereas the Inviscid Burgers equation requires appropriate inflow transport boundary conditions. With dimensional variables, $u$ has units of velocity and $\nu$ of kinematic viscosity; all numerical examples below use dimensionless quantities.

The Burgers' equation with constant linear damping has the displayed form. For zero viscosity and $\gamma >0$, characteristics are $u=u_0(a)e^{-\gamma t}$ and $x=a+u_0(a)(1-e^{-\gamma t})/\gamma$. A negative initial slope can make this coordinate map cease to be invertible, producing shock formation by characteristic intersection. The transformation $u=e^{-\gamma t}v$, $\tau =(1-e^{-\gamma t})/\gamma$ converts the inviscid equation to the ordinary inviscid Burgers' equation. Across a thin viscous layer the damping contributes only at lower order to the leading Rankine-Hugoniot condition.

The cubic-flux variant replaces the quadratic conservative flux of Burgers' equation by $-q^3/3$. Its inviscid characteristic speed is $-q^2$, while positive diffusivity regularizes shock waves. It is not linearized by the usual Cole-Hopf transformation.

Write $L=\log \psi$, with $\psi >0$. Under the Cole-Hopf transformation $q=2ϵL_{\theta }$, direct differentiation givesSince $L_{\theta \theta }+L_{\theta }^2={\psi }_{\theta \theta }/\psi$, Burgers' equation is equivalent toThe ratio is therefore a function $a(Z)$ alone. Multiplying $\psi$ by $\exp [-\int a(Z)dZ]$ leaves $q$ unchanged and removes this scalar freedom. Thus one may choose the normalization for whichThis is the heat equation. The positive sign in $q=2ϵ{\partial }_{\theta }\log \psi$ corresponds to the negative conservative flux $-q^2/2$ used here; the more usual positive-flux convention has the opposite sign.