Does the complex deformation of the Riemann equation exhibit shocks?

Carl M. Bender; Joshua Feinberg

doi:10.1088/1751-8113/41/24/244004

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arXiv · 0709.2727

Does the complex deformation of the Riemann equation exhibit shocks?

Carl M. Bender·Joshua Feinberg

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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.

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Carl M. Bender, Joshua Feinberg. 2007-09-17. Does the complex deformation of the Riemann equation exhibit shocks?. https://doi.org/10.1088/1751-8113/41/24/244004

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