Samokhin A. Numeric simulation of sawtooth solutions of the Burgers equation on a finite interval, talk at The Workshop on Integrable Nonlinear Equations, 18-24 October 2015, Mikulov, Czech Republic (abstract)

Speaker: Alexey Samokhin

Title: Numeric simulation of sawtooth solutions of the Burgers equation on a finite interval

Abstract:
Properties of the solutions to the Burgers equation $u_t={\epsilon }^2{u}_{xx}-2u{u}_x$ on a finite interval $x\in [0,L]$ are studied. The initial value/boundary conditions model a periodic perturbation on the left boundary:

$u(x,0)=a,u(0,t)=a+b\sin (\omega t),u_x(L,t)=0$

The asymptotics of the solution for this problem at $L\to \mathrm{\infty }$ coincides with the well known Fay solution

$u=\frac{a}{R}\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (n\theta )}{\sinh (n(1+X)/2\cdot R}$

here $R$ is the Reynolds number, $\theta =\omega (t-x/u_0)$.

In particular, $\underset{x\to +\mathrm{\infty }}{\lim }u(x,t)=a$, which is the solution's average value over $x>0$.

Not so for another asymptotics, at $t\to +\mathrm{\infty }$. The form of the solution retains the sawtooth profile yet its average over $[0,L]$ differs from $a$ and depends also on the perturbation amplitude $b$. Interaction between two perturbations of different frequencies is discussed.