Asymptotics of the solution of a boundary value problem for transverse nonlinear electromagnetic wave

V. A. Getman^a, T. F. Dolgikh^a, M. Yu. Zhukov<sup>ab

a</sup> Southern Federal University, 8 a Milchakov St., Rostov-on-Don 344090, Russia
^b Southern Mathematical Institute VSC RAS, 53 Vatutin St., Vladikavkaz 362025, Russia

Abstract: An asymptotic solution of the boundary value problem for two quasi-linear hyperbolic equations describing the behavior of a transverse electromagnetic wave (TEM wave) in a nonlinear continuous medium is constructed when the dependence of polarization $P$ on the electric field strength $E$ (physical nonlinearity) has the form $P =\varepsilon_0(\chi_1 E + \chi_2 E^2 + \chi_3 E^3)$, where $\chi_1$, $\chi_2$, $\chi_3$ are dielectric susceptibility and $\varepsilon_0$ is the dielectric constant of vacuum. The main term of the asymptotics is constructed in two cases: (i) $\chi_1 = O(1)$, $\chi_2\to 0$, $\chi_3 =0$ (anisotropic continuous medium), (ii) $\chi_1=O(1)$, $\chi_2= 0$, $\chi_3\to 0$ (isotropic continuous medium), although one of the methods used to construct the asymptotics is easily transferred to the case (iii) $\chi_1= O(1)$, $\chi_2\to 0$, $\chi_3\to 0$. In the case of (i), the asymptotics for $\chi_2\to 0$ is constructed in two ways. In the first variant, the direct expansion in a series by a small parameter of the exact implicit solution of the boundary value problem is used with the subsequent numerical construction of the explicit solution on the lines of the level of the implicit solution (the main term of the asymptotics of the implicit solution). In the second variant, the expansion into series by parameter is carried out at all stages preceding the construction of an exact implicit solution, which leads to an implicit solution different from the exact one, but the main term of the asymptotics of the new and previous solutions coincide. The equivalence of these two options is far from obvious, in particular, the exact implicit solution contains the hypergeometric Gauss function, and the asymptotic implicit solution contains the Bessel function. In the case of (ii), the asymptotics at $\chi_3\to 0$ can be constructed only in the second way, by performing parameter decomposition at all stages of constructing an implicit solution. The first variant of constructing the asymptotics is indispensable to them, due to the fact that an exact implicit solution cannot be constructed. The hodograph method, based on the conservation law for a system of two quasi-linear hyperbolic equations of type $1 + 1$ in partial derivatives of the first order, was used to construct a solution to the problem of the behavior of TEM waves , both exact and asymptotic. The method allows to transform a system of quasi-linear equations into one linear partial differential equation of the second order with variable coefficients. The effectiveness of the method depends on the presence of explicit relations connecting the initial variables with the Riemann invariants, as well as on the presence of an explicit expression for the Riemann–Green function of a linear differential equation. In cases (i), (ii) the specified conditions are valid. The presented results allow us to trace in detail the evolution of TEM waves in nonlinear media, for example, in coaxial waveguides or distributed ideal transmission lines, in particular, to determine the time (and spatial coordinate) at which the occurrence of shock electromagnetic waves is possible.

Key words: systems of quasi-linear hyperbolic equations, Riemann invariants, Riemann–Green function, hodograph method, asymptotic expansions.

UDC: 514.743.48, 517.956.35, 517.955.8

MSC: 35F15, 35L40, 41A58

Received: 14.12.2024

DOI: 10.46698/e7486-7095-0322-l