Burgers' equation

Viscous Burgers' equation

The viscous Burgers' equation can be converted to a linear equation by the Cole–Hopf transformation,

${\displaystyle u(x,t)=-2\nu \frac{\mathrm{\partial }}{\mathrm{\partial }x}\ln \phi (x,t),}$

which turns it into the equation

${\displaystyle 2\nu \frac{\mathrm{\partial }}{\mathrm{\partial }x}\left[\frac{1}{\phi }\left(\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}-\nu \frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{x}^2}\right)\right]=0,}$

which can be integrated with respect to ${\displaystyle x}$ to obtain

${\displaystyle \frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}-\nu \frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{x}^2}=\phi \frac{df(t)}{dt},}$

where ${\displaystyle df/dt}$ is an arbitrary function of time. Introducing the transformation ${\displaystyle \phi \to \phi e^f}$ (which does not affect the function ${\displaystyle u(x,t)}$), the required equation reduces to that of the heat equation

${\displaystyle \frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}=\nu \frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{x}^2}.}$

The diffusion equation can be solved. That is, if ${\displaystyle \phi (x,0)={\phi }_0(x)}$, then

${\displaystyle \phi (x,t)=\frac{1}{\sqrt{4\pi \nu t}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\phi }_0(x^{\prime })\exp \left[-\frac{(x-x^{\prime }{)}^2}{4\nu t}\right]d{x}^{\prime }.}$

The initial function ${\displaystyle {\phi }_0(x)}$ is related to the initial function ${\displaystyle u(x,0)=f(x)}$ by

${\displaystyle \ln {\phi }_0(x)=-\frac{1}{2\nu }{\int }_0^{x}f(x^{\prime })d{x}^{\prime },}$

where the lower limit is chosen arbitrarily. Inverting the Cole–Hopf transformation, we have

${\displaystyle u(x,t)=-2\nu \frac{\mathrm{\partial }}{\mathrm{\partial }x}\ln \left\{\frac{1}{\sqrt{4\pi \nu t}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left[-\frac{(x-x^{\prime }{)}^2}{4\nu t}-\frac{1}{2\nu }{\int }_0^{x^{\prime }}f(x^{″})d{x}^{″}\right]d{x}^{\prime }\right\}}$

which simplifies, by getting rid of the time-dependent prefactor in the argument of the logarithm, to

${\displaystyle u(x,t)=-2\nu \frac{\mathrm{\partial }}{\mathrm{\partial }x}\ln \left\{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left[-\frac{(x-x^{\prime }{)}^2}{4\nu t}-\frac{1}{2\nu }{\int }_0^{x^{\prime }}f(x^{″})d{x}^{″}\right]d{x}^{\prime }\right\}.}$

This solution is derived from the solution of the heat equation for ${\displaystyle \phi }$ that decays to zero as ${\displaystyle x\to \pm \mathrm{\infty }}$; other solutions for ${\displaystyle u}$ can be obtained starting from solutions of ${\displaystyle \phi }$ that satisfies different boundary conditions.

Some explicit solutions of the viscous Burgers' equation

Explicit expressions for the viscous Burgers' equation are available. Some of the physically relevant solutions are given below:

Steadily propagating traveling wave

If ${\displaystyle u(x,0)=f(x)}$ is such that ${\displaystyle f(-\mathrm{\infty })=f^{+}}$ and ${\displaystyle f(+\mathrm{\infty })=f^{-}}$ and ${\displaystyle f^{\prime }(x)<0}$, then we have a traveling-wave solution (with a constant speed ${\displaystyle c=(f^{+}+f^{-})/2}$) given by

${\displaystyle u(x,t)=c-\frac{f^{+}-f^{-}}{2}\tanh \left[\frac{f^{+}-f^{-}}{4\nu }(x-ct)\right].}$

This solution, that was originally derived by Harry Bateman in 1915, is used to describe the variation of pressure across a weak shock wave. When ${\displaystyle f^{+}=2}$ and ${\displaystyle f^{-}=0}$ this simplifies to

${\displaystyle u(x,t)=\frac{2}{1+e^{\frac{x-t}{\nu }}}}$

with ${\displaystyle c=1}$.

Delta function as an initial condition

If ${\displaystyle u(x,0)=2\nu \mathrm{R}\mathrm{e}\delta (x)}$, where ${\displaystyle \mathrm{R}\mathrm{e}}$ (say, the Reynolds number) is a constant, then we have

${\displaystyle u(x,t)=\sqrt{\frac{\nu }{\pi t}}\left[\frac{(e^{\mathrm{R}\mathrm{e}}-1)e^{-x^2/4\nu t}}{1+(e^{\mathrm{R}\mathrm{e}}-1)\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(x/\sqrt{4\nu t})/2}\right].}$

In the limit ${\displaystyle \mathrm{R}\mathrm{e}\to 0}$, the limiting behaviour is a diffusional spreading of a source and therefore is given by

${\displaystyle u(x,t)=\frac{2\nu \mathrm{R}\mathrm{e}}{\sqrt{4\pi \nu t}}\exp \left(-\frac{x^2}{4\nu t}\right).}$

On the other hand, In the limit ${\displaystyle \mathrm{R}\mathrm{e}\to \mathrm{\infty }}$, the solution approaches that of the aforementioned Chandrasekhar's shock-wave solution of the inviscid Burgers' equation and is given by

${\displaystyle u(x,t)=\{\begin{array}{l}\frac{x}{t},0<x<\sqrt{2\nu \mathrm{R}\mathrm{e}t},\\ 0,\text{otherwise}.\end{array}}$

The shock wave location and its speed are given by ${\displaystyle x=\sqrt{2\nu \mathrm{R}\mathrm{e}t}}$ and ${\displaystyle \sqrt{\nu \mathrm{R}\mathrm{e}/t}.}$

N-wave solution

The N-wave solution comprises a compression wave followed by a rarefaction wave. A solution of this type is given by

${\displaystyle u(x,t)=\frac{x}{t}{\left[1+\frac{1}{e^{{\mathrm{R}\mathrm{e}}_0-1}}\sqrt{\frac{t}{t_0}}\exp \left(-\frac{\mathrm{R}\mathrm{e}(t)x^2}{4\nu {\mathrm{R}\mathrm{e}}_{0}t}\right)\right]}^{-1}}$

where ${\displaystyle {\mathrm{R}\mathrm{e}}_0}$ may be regarded as an initial Reynolds number at time ${\displaystyle t=t_0}$ and ${\displaystyle \mathrm{R}\mathrm{e}(t)=(1/2\nu ){\int }_0^{\mathrm{\infty }}udx=\ln (1+\sqrt{\tau /t})}$ with ${\displaystyle \tau =t_0\sqrt{e^{{\mathrm{R}\mathrm{e}}_0}-1}}$, may be regarded as the time-varying Reynolds number.

Other forms

Multi-dimensional Burgers' equation

In two or more dimensions, the Burgers' equation becomes

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\cdot \mathrm{\nabla }u=\nu {\mathrm{\nabla }}^{2}u.}$

One can also extend the equation for the vector field ${\displaystyle \mathbf{u}}$, as in

${\displaystyle \frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}=\nu {\mathrm{\nabla }}^2\mathbf{u}.}$

Generalized Burgers' equation

The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e.,

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+c(u)\frac{\mathrm{\partial }u}{\mathrm{\partial }x}=\nu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}.}$

where ${\displaystyle c(u)}$ is any arbitrary function of u. The inviscid ${\displaystyle \nu =0}$ equation is still a quasilinear hyperbolic equation for ${\displaystyle c(u)>0}$ and its solution can be constructed using method of characteristics as before.

Stochastic Burgers' equation

Added space-time noise ${\displaystyle \eta (x,t)=\dot{W}(x,t)}$, where ${\displaystyle W}$ is an ${\displaystyle L^2(\mathbb{R})}$ Wiener process, forms a stochastic Burgers' equation

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}=\nu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}-\lambda \frac{\mathrm{\partial }\eta }{\mathrm{\partial }x}.}$

This stochastic PDE is the one-dimensional version of Kardar–Parisi–Zhang equation in a field ${\displaystyle h(x,t)}$ upon substituting ${\displaystyle u(x,t)=-\lambda \mathrm{\partial }h/\mathrm{\partial }x}$.