Does the complex deformation of the Riemann equation exhibit shocks?

TL;DR

This paper investigates a complex deformation of the Riemann equation, revealing that shocks do not form unless the deformation parameter is an odd integer, thus extending understanding of shock formation in complexified fluid models.

Contribution

The paper introduces an exact solution to a complex deformation of the Riemann equation and identifies conditions under which shocks can or cannot develop.

Findings

Shocks do not develop for real initial conditions unless the deformation parameter is an odd integer.

The complex deformation is exactly solvable using the method of characteristic strips.

The deformation preserves $ ext{PT}$ symmetry and alters shock formation criteria.

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.