Does the complex deformation of the Riemann equation exhibit shocks?

Abstract

The Riemann equation ut + uux = 0, which describes a one-dimensional accelerationless perfect fluid, possesses solutions that typically develop shocks in a finite time. This equation is symmetric. A one-parameter -invariant complex deformation of this equation, ut − iu(iux)ϵ = 0 (ϵ real), is solved exactly using the method of characteristic strips, and it is shown that for real initial conditions, shocks cannot develop unless ϵ is an odd integer. When ϵ is an odd integer, the shock-formation time is calculated explicitly.