K.M. Zuev 1, N.A. Kudryashov 2
Department ≪Applied Mathematics≫, Institute of Laser and Plasma Technologies, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), RU-115409 Moscow, Russia
Abstract: In this paper it is investigated the generalized Vakhnenko-Parkes equation describing the propagation of short-wavelength disturbances in relaxing media with amplitude-dependent wave velocity. As the Cauchy problem for this equation proves intractable to the inverse scattering transform method, the analysis employs traveling wave variables. To study the integrability of an ordinary differential equation obtained by moving to traveling wave variables, the Painlev´e test is used. A general solution to this ordinary differential equation, written in terms of quadrature, is obtained. The dependence of the solution on parameters of the equation is investigated. Based on the analysis, explicit exact solutions in traveling wave variables are found. It is shown that the equation has periodic solutions expressed through Jacobi elliptic functions, as well as an singular analytical solution in explicit form. Obtained exact solutions can be used as test functions in the analysis of results of numerical modeling of processes in relaxing media described by equations of the Vakhnenko-Parkes type.
© 2026 European Society of Computational Methods in Sciences and Engineering
Keywords: Vakhnenko-Parkes equation, Painlev´e analysis, exact solutions, solutions in quadratures, traveling wave solutions, Jacobi elliptic functions
Mathematics Subject Classification: 35Q51, 34M55, 35C07
PACS: 02.30.Jr, 47.35.Fg, 02.30.Hq, 47.35.Jk