PT-symmetrically deformed shock waves

TL;DR

This paper explores how PT-symmetrical deformation transforms shock waves into peaked solutions with derivative discontinuities across a broad class of nonlinear wave equations, including the inviscid Burgers equation, revealing new wave behaviors.

Contribution

It introduces a generalized PT-deformation map applicable to various nonlinear wave equations and demonstrates the transformation of shocks into peaks, expanding understanding of wave behavior under PT-symmetry.

Findings

Real solutions become peaked with derivative discontinuities after PT-deformation.

The PT-deformed inviscid Burgers equation does not produce shocks but peaks instead.

Complex solutions under PT-deformation develop discontinuities.

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.