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Eulerian calculations of wave breaking and multivalued solutions in a traveling wave tube

The traveling wave tube is an electron beam device that works on a similar principle to the beam-plasma instability, where the background plasma is replaced by an electromagnetic waveguiding structure. The nonlinear evolution of the instability includes wave breaking and the formation of multivalued solutions, and conventionally these solutions have been computed using Lagrangian techniques. Recently, an Eulerian method for computing multivalued solutions was developed in the context of geometrical optics and has been applied to the klystron, a relative of the traveling wave tube. In this paper we apply the Eulerian technique to solve a traveling wave tube model and compare the results to a Lagrangian technique. The results are found to be in good qualitative agreement with small quantitative differences that are attributed to the numerical methods.