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PT-symmetrically deformed shock waves

Cavaglia, A. and Fring, A. (2012). PT-symmetrically deformed shock waves. Journal of Physics A: Mathematical and Theoretical, 45(44), p. 444010. doi: 10.1088/1751-8113/45/44/444010

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 ε. The peaks are formed as a result of mapping the multi-valued self-avoiding shock profile to a multivalued 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.

Publication Type: Article
Additional Information: Copyright IOP publishing, 2012. This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1751-8113/45/44/444010.
Subjects: Q Science > QA Mathematics
Q Science > QC Physics
Departments: School of Mathematics, Computer Science & Engineering > Mathematics
URI: http://openaccess.city.ac.uk/id/eprint/13891
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