Does the complex deformation of the Riemann equation exhibit shocks?

Details Does the complex deformation of the Riemann equation exhibit shocks? Does the complex deformation of the Riemann equation exhibit shocks?

2007-09-17

arxiv.org/abs/0709.2727v1

J.Phys.A41:244004,2008

Physics

High Energy Physics

High Energy Physics - Theory

latex, 8 pages

Scientific paper

10.1088/1751-8113/41/24/244004

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.

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