# 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 ${\displaystyle u_{t}=\varepsilon ^{2}u_{xx}-2uu_{x}}$ on a finite interval ${\displaystyle x\in [0,\,L]}$ are studied. The initial value/boundary conditions model a periodic perturbation on the left boundary:

${\displaystyle u(x,0)=a,\quad u(0,t)=a+b\sin(\omega t),\quad u_{x}(L,t)=0}$

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

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

here ${\displaystyle R}$ is the Reynolds number, ${\displaystyle \theta =\omega (t-x/u_{0})}$.

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

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

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