= Viscous Burgers travelling shock
{c}
{title2=$U(z)=s-\frac{u_L-u_R}{2}\tanh\frac{(u_L-u_R)z}{4\nu}$}
A <travelling wave> of the <viscous Burgers equation> can connect constant states $u_L>u_R$ through a smooth decreasing layer. Require $u(x,t)=U(z)$, $z=x-st$, $U(-\infty)=u_L$, $U(+\infty)=u_R$, and $U'\to0$ at both ends. Substitution and one <integration> give
$$
(U-s)U'=\nu U'',\qquad \nu U'=\frac12(U-u_L)(U-u_R),\qquad s=\frac{u_L+u_R}{2}.
$$
The speed follows by evaluating the integration constant at both end states. It is precisely the <Rankine-Hugoniot condition> for <conservation law flux> $f(u)=u^2/2$. Choosing the center $U(0)=s$ gives the displayed <hyperbolic tangent> profile. Its characteristic width scales as $\nu/(u_L-u_R)$; a smaller positive $\nu$ makes a thinner smooth layer.
For an original worked example in <dimensionless quantities>, take $u_L=2$, $u_R=0$ and $s=1$:
$$
u(x,t)=1-\tanh\frac{x-t}{2\nu}=\frac{2}{1+\exp((x-t)/\nu)}.
$$
This is an exact <classical solution> on $\mathbb R$ with its own smooth <initial condition> $u(x,0)=1-\tanh(x/(2\nu))$ and limiting states $2,0$. It is a family of moving profiles, not the finite-$\nu$ solution starting from an exact discontinuous step. For that different <initial condition>, the <Cole-Hopf transformation> supplies the evolution. At $\nu=1/2$, this front starts from the same $1-\tanh x$ profile as the <Burgers characteristic breaking> example: positive <diffusion> maintains a smooth moving layer while the inviscid profile breaks at $t=1$.
To check the <partial differential equation> explicitly, set $w=(x-t)/(2\nu)$ and $S=1-\tanh^2w$. Then
$$
u_t=\frac{S}{2\nu},\qquad u_x=-\frac{S}{2\nu},\qquad u_{xx}=\frac{S\tanh w}{2\nu^2},\qquad u_t+uu_x=\frac{S\tanh w}{2\nu}=\nu u_{xx}.
$$
The independent <Rankine-Hugoniot condition> check gives $s=(0-2)/(0-2)=1$, using $f(2)=2$ and $f(0)=0$. At fixed $x\ne t$, the <vanishing viscosity approximation> tends to the <entropy shock> $2$ for $x<t$ and $0$ for $x>t$. At the center the smooth profiles stay at $1$; changing the value on the discontinuity does not change the limiting <weak solution>.
\Image[/burgers-viscous-travelling-shock.png]
{title=Exact viscous fronts from 2 to 0: the speed stays 1 while the layer narrows with viscosity}
For $u_L<u_R$, the same integrated <ordinary differential equation> has $U'<0$ between the two states when $\nu>0$, so it cannot supply an increasing connecting front. The inviscid admissible expansion is the <rarefaction wave> of the <Burgers Riemann problem with positive flux>.
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