Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2686710.22363/2658-4670-2021-29-2-114-125Research ArticleInvestigation of the existence domain for Dyakonov surface waves in the Sage computer algebra systemKroytorOleg K.<p>Postgraduate of Department of Applied Probability and Informatics</p>kroytor_ok@pfur.ruhttps://orcid.org/0000-0002-5691-7331Peoples’ Friendship University of Russia (RUDN University)2806202129211412528062021Copyright © 2021, Kroytor O.K.2021<p style="text-align: justify;">Surface electromagnetic waves (Dyakonov waves) propagating along a plane interface between an isotropic substance with a constant dielectric constant and an anisotropic crystal, whose dielectric tensor has a symmetry axis directed along the interface, are considered. It is well known that the question of the existence of such surface waves is reduced to the question of the existence of a solution to a certain system of algebraic equations and inequalities. In the present work, this system is investigated in the Sage computer algebra system. The built-in technique of exceptional ideals in Sage made it possible to describe the solution of a system of algebraic equations parametrically using a single parameter, with all the original quantities expressed in terms of this parameter using radicals. The remaining inequalities were only partially investigated analytically. For a complete study of the solvability of the system of equations and inequalities, a symbolic-numerical algorithm is proposed and implemented in Sage, and the results of computer experiments are presented. Based on these results, conclusions were drawn that require further theoretical substantiation.</p>surface wavesDyakonov waveselectromagnetic wavescomputer algebraSageповерхностные волныволны Дьяконоваэлектромагнитные волныкомпьютерная алгебраSage<p>1. Introduction In the 1980s, a special class of solutions to Maxwells equations was theoret- ically discovered, namely, electromagnetic waves traveling along the interface between two dielectrics, the intensity of which rapidly decreases with distance from the interface [1]-[6]. These waves are called Dyakonov surface waves. Experimental observation of surface waves was carried out quite recently [7], [8]. In theoretical works, as in the work of Dyakonov itself [2], the question of the existence of surface waves was reduced to the question of the existence of a solution to a certain system of algebraic equations and inequalities that cannot be solved analytically, which hinders further research. In this paper, it will be shown what computer algebra systems can give for these systems. 2. Surface waves We investigate the classical problem of waves propagating along the interface of an anisotropic medium with permittivity</p>[F. N. Marchevskii, V. L. Strizhevskii, and S. V. Strizhevskii, “Singular electromagnetic waves in bounded anisotropic media,” Sov. Phys. Solid State, vol. 26, p. 857, 1984.][M. I. Dyakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP, vol. 67, p. 714, 1988.][O. Takayama, L.-C. Crasovan, S. Johansen, D. Mihalache, D. Artigas, and L. Torner, “Dyakonov surface waves: a review,” Electromagnetics, vol. 28, pp. 126-145, 2008. DOI: 10.1080/02726340801921403.][J. A. Polo Jr. and A. Lakhtakia, “A Surface Electromagnetic Waves: a Review,” Laser & Photonics Reviews, vol. 5, pp. 234-246, 2011. DOI: 10.1002/lpor.200900050.][O. N. Bikeev and L. A. Sevastianov, “Surface electromagnetic waves at the interface of two anisotropic media,” RUDN Journal of Mathematics, Information Sciences and Physics, vol. 25, no. 2, pp. 141-148, 2017, in Russian. DOI: 10.22363/2312-9735-2017-25-2-141-148.][O. N. Bikeev, K. P. Lovetskiy, N. E. Nikolaev, L. A. Sevastianov, and A. A. Tiutiunnik, “Electromagnetic surface waves guided by a twist discontinuity in a uniaxial dielectric with optic axis lying in the discontinuity plane,” Journal of Electromagnetic Waves and Applications, vol. 33, no. 15, pp. 2009-2021, 2017. DOI: 10.1080/09205071.2019.1655486.][O. Takayama, L.-C. Crasovan, D. Artigas, and L. Torner, “Observation of Dyakonov surface waves,” Physical Review Letters, vol. 102, p. 043-903, 2009. DOI: 10.1103/PhysRevLett.102.043903.][O. Takayama, D. Artigas, and L. Torner, “Lossless directional guiding of light in dielectric nanosheets using Dyakonov surface waves,” Nature Nanotech, vol. 9, pp. 419-424, 2014. DOI: 10.1038/nnano.2014.90.][D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms, 2nd ed. Springer, 1997.]