Does the complex deformation of the Riemann equation exhibit shocks?

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, $u_t-iu(iu_x)^ε= 0$ ($ε$ 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.

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