Cole-Hopf transformation ($u=-2\nu {\partial }_x\log \varphi$)

The Cole-Hopf transformation $u=-2\nu {\varphi }_x/\varphi$ converts the viscous Burgers equation into the heat equation ${\varphi }_t=\nu {\varphi }_{xx}$. Initial data $u(x,0)=g(x)$ correspond to $\varphi (x,0)=C\exp [-(2\nu {)}^{-1}{\int }^{x}g(s)ds]$.

Substituting $u=-2\nu {\partial }_x\log \Psi$ into the viscous Burgers equation gives ${\partial }_x[({\Psi }_t-\nu {\Psi }_{xx})/\Psi ]=0$. Integration permits the arbitrary function $C(t)$ in the title. Multiplication of $\Psi$ by $e^{-\int C(t)dt}$ preserves $u$ and reduces its equation to the heat equation. Thus the heat-equation representation uses a permissible time normalization rather than a condition uniquely imposed on every representative $\Psi$.

For $q_Z-q{q}_{\theta }=\delta q_{\theta \theta }$ and initial data $q=\sin \theta$, the Cole-Hopf transformation $q=2\delta {\psi }_{\theta }/\psi$ gives $\psi (\theta ,0)=\exp [-\cos \theta /(2\delta )]$. The Fourier series of this exponential and the heat equation give the displayed exact positive diffusion solution. Its first harmonic gives $q\sim 4\delta [I_1(a)/I_0(a)]e^{-\delta Z}\sin \theta$. The large-argument asymptotic expansion of a modified Bessel function makes the ratio tend to one as $\delta \to 0$.

For the negative-flux viscous Burgers equation $q_Z-q{q}_{\theta }=ϵq_{\theta \theta }$, set $q=2ϵ{\partial }_{\theta }\log \psi$. A sum of exponentials in the initial heat data evolves into $\psi ={\sum }_j{e}^{A_j}$, giving the displayed softmax function average. Two terms produce a traveling tanh shock from the lower to the higher state, with speed minus their mean.

For ordered states $q_1<\cdots <q_N$, in the moving frame $\theta =sZ+\xi$ the first and last exponents in the finite-exponential Cole-Hopf solution have the same leading coefficient. Each interior state's coefficient is smaller by $(q_j-q_1)(q_N-q_j)/(4ϵ)>0$. Thus the large-$Z$ moving-frame profile is the two-state shock between the extremes. At fixed $\theta$, the endpoint of largest absolute value dominates; equal opposite extremes leave a stationary shock.