PT-symmetrically deformed shock waves

Abstract

We investigate for a large class of nonlinear wave equations, which allow for shock wave formations, how these solutions behave when they are PT-symmetrically deformed. For real solutions we find that they are transformed into peaked solutions with a discontinuity in the first derivative instead. The systems we investigate include the PT-symmetrically deformed inviscid Burgers equation recently studied by Bender and Feinberg, for which we show that it does not develop any shocks, but peaks instead. In this case we exploit the rare fact that the PT-deformation can be provided by an explicit map found by Curtright and Fairlie together with the property that the undeformed equation can be solved by the method of characteristics. We generalise the map and observe this type of behaviour for all integer values of the deformation parameter epsilon. The peaks are formed as a result of mapping the multi-valued self-avoiding shock profile to a multi-valued self-crossing function by means of the PT-deformation. For some deformation parameters we also investigate the deformation of complex solutions and demonstrate that in this case the deformation mechanism leads to discontinuties.

17 pages, 10 figures

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Cited by in corpus (6)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Perfectly invisible $\mathcal{P}\mathcal{T}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry
• Complex solitons with real energies
• Time-delay and reality conditions for complex solitons
• PT-symmetry in quasi-integrable models
• Isomorphic Hilbert spaces associated with different Complex Contours of the $\mathcal{P}\mathcal{T}$-Symmetric $(-x^4)$ Theory