Does the complex deformation of the Riemann equation exhibit shocks?

Abstract

The Riemann equation $u_t+uu_x=0$, which describes a one-dimensional accelerationless perfect fluid, possesses solutions that typically develop shocks in a finite time. This equation is $\cP\cT$ symmetric. A one-parameter $\cP\cT$-invariant complex deformation of this equation, $u_t-iu(iu_x)^\epsilon= 0$ ($\epsilon$ real), is solved exactly using the method of characteristic strips, and it is shown that for real initial conditions, shocks cannot develop unless $\epsilon$ is an odd integer.