Shock Layer Movement for Burgers' equation

This paper considers the motion of the shock layer exhibited by the solution of the initial-boundary value problem for the equation $\epsilon u_{xx} + uu_x = u_t $ with small values of $\epsilon > 0$. This Burgers’ equation occurs as a model for a number of physical problems and is representative of many convection-dominated evolution equations. A specific example will be solved explicitly using the Cole–Hopf transformation. The result suggests a corresponding asymptotic approach which provides the shock layer location $x_\epsilon ( t )$ as the solution of an initial value problem. That method, which can be applied to a far wider class of equations, also shows how the steady-state shock location can be changed substantially by asymptotic exponentially small perturbations.