ON THE PROBLEM OF NONLINEAR WAVES IN RODS

Abstract

For the English abstract and full text of the article please see the attached PDF-File (English version follows Russian version).ABSTRACT The article shows new, nonclassical dispersion waves and vibrations, and their spectrum is studied for two reasons: firstly, there is an increase in the number of composite materials used in railway transport, which leads to the phenomenon of dispersion, and secondly, the spring of 2017 showed that many kilometers of tracks were flooded with shallow water. The task of detecting a track defect and determining its spatial localization can be solved with the help of measuring the energy of longitudinal, vertical and transverse oscillations. But the phenomenon of dispersion makes it very difficult to detect the occurrence of new track defects from the rolling stock (see World of Transport and Transportation, 2016, Vol. 14, Iss. 3). The method of «unfixed constructive change of variables» is used in the paper. The authors conclude that two necessary conditions for the solvability of a system of differential equations with first-order partial derivatives have one common nontrivial factor; and that through designing new exact solutions of the non-linear KdV equation that differ from the classical ones, it is found that there is a change of variables in which the equation for the function Y(ξ, d) = Y0(U(ξ, d), ξ) is separate from all the others. Keywords: railway track, vibration spectrum, diagnostic system, Korteweg-de Vries equations, method of non-fixed constructive change of variables. Acknowledgements. The authors are grateful to Volosov K. A., Danilov V. G., Zaitsev V. F., Kudryashov A. N., Karasev M. V., Maslov V. P., Novikov R. G., Shefarevich A. I. for useful discussions and criticisms.