We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameter h 1. This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intensity lies. Because of the matrix character of the Dirac equation, this wavefunction contains a nontrivial semiclassical phase, which is absent for a scalar wave equation and which influences the focusing. We carefully discuss the three steps in our semiclassical approach: the adiabatic reduction of the matrix equation to an effective scalar equation, the construction of the wavefunction using the Maslov canonical operator and the application of the uniform approximation to the integral expression for the wavefunction in the vicinity of a caustic. We consider several numerical examples and show that our semiclassical results are in very good agreement with the results of tight-binding calculations. In particular, we show that the semiclassical phase can have a pronounced effect on the position of the focus and its intensity.

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We consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for the nonlinear Klein-Gordon equation and for the Korteweg-de Vries equation. In this case, the leading asymptotic expansion term has the form X (S(x, t)1 is a small parameter and the phase S(x, t) and slowly changing parameters I(x, t) are to be found from the system of "averaged" Whitham equations. We obtain the equations for the phase shift Φ(x, t) by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. Our observation, which essentially follows from papers by Haberman and collaborators, is that if we incorporate the phase shift Φ into the phase and adjust the parameterĨ by setting (x, t, h) become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term, which is The nonlinear Klein-Gordon equationWe consider the Cauchy problem for the nonlinear Klein-Gordon equationHere, V (u, x, t), c(x, t), u 0 (x), and v 0 (x) ∈ R are smooth functions, t ∈ R, x ∈ R n , and 0 < h 1 is a small parameter. We impose the condition on V (u, x, t) that ensures the existence of rapidly oscillating real-valued asymptotic solutions of Eq. (1.1) [1]: there exist smooth functions a(x, t) < b(x, t) such that for every fixed x and t in the interval a < E < b, the equation V (u, x, t) = E has at least two solutions u min (x, t, E) and u max (x, t, E), u min < u max , that depend smoothly on x, t, and E and are such thatThe potential V and the initial conditions u 0 and v 0 can depend on the real parameter (or parameters) ε. For example, we can consider the potential V (u, x, t, ε) = 1 2 ω(x, t)u 2 + εV 1 (u, x, t), V 1 (u, x, t) = O(u 3 ).

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