Sonic shocks governed by the modified Burgers' equation

Abstract

In this paper, we investigate the evolution of N-waves in a medium governed by the modified Burgers' equation. It is shown that the general behaviour when the nonlinearity is of arbitrary odd integer order is the same as for the cubic case. For an N-wave of zero mean displacement, a shock is formed immediately to prevent a multi-valued solution and a second shock is formed at later times. At a finite time, the second shock satisfies a sonic condition and this state persists. The Taylor-type shock structure ceases to be the appropriate description, and instead we have a shock which matches only algebraically to the outer wave on one side. At a larger time still, the other shock is affected but the two shocks remain distinct until the wave dies under linear mechanisms. The behaviour of N-waves of non-zero mean is also examined and it is shown that in some cases, a purely one-signed profile remains.