Does the complex deformation of the Riemann equation exhibit shocks?

Abstract

The Riemann equation $u_t+u{u}_x=0$, 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, $u_t-iu(i{u}_x{)}^{\epsilon }=0$ ($\epsilon$ 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.

latex, 8 pages

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• PT-Symmetric Extension of the Korteweg-de Vries Equation
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Cited by in corpus (11)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Chaotic systems in complex phase space
• PT-symmetry breaking in complex nonlinear wave equations and their deformations
• Perfectly invisible $\mathcal{P}\mathcal{T}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry
• Probability Density in the Complex Plane
• Time-delay and reality conditions for complex solitons
• From real fields to complex Calogero particles
• Periodic orbits for classical particles having complex energy
• Conduction bands in classical periodic potentials
• PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation
• Integrable models from PT-symmetric deformations