Andrea Cavaglia, Andreas Fring
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.
View original:
http://arxiv.org/abs/1201.5809
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