Nonlinear models of wave processes in the dimension (1 + 1) and first order quasilinear equations 

Viktor M. Zhuravlev, Doctor of physical and mathematical sciences, professor, researcher of the laboratory of gravity, cosmology and astrophysics, Ulyanovsk State Pedagogical University (4/5 Lenin square, Ulyanovsk, Russia), E-mail: zhvictorm@gmail.com 

Background. The study considers the connection of models of nonlinear wave processes in various one-dimensional physical systems, such as nonlinear two-wire lines, with first order quasilinear equations. The relation established in this work makes it possible to obtain exact solutions of a large class of nonlinear wave equations of the second order, for which one arbitrary initial or boundary condition can be set. Equations of this type and their solutions play an important role in many problems of electrodynamics and acoustics. Materials and methods. The main method used in the work is the method of characteristics for constructing solutions of first order quasilinear equations. Their connection with nonlinear wave equations of the second order is established in the form of simple differential relations. Results. The study presents a classification of second order nonlinear wave equations based on the classification of first-order quasilinear equations according to the functional form of the coefficients of these equations. Exact solutions of the equations under consideration, including the initial problem with one arbitrary initial condition, are given. The proposed approach is applied to the derivation of partially integrable equations of non-linear two-wire lines associated with first order quasilinear equations. The most interesting types of non-linear two-wire lines with volt-ampere and capacitance-voltage characteristics of their linear elements are given from the point of view of practical application. Conclusions. The developed approach makes it possible to construct exact solutions of a set of wave nonlinear equations based on exact solutions of first order quasilinear equations. 

nonlinear wave equations, first order quasilinear equations, nonlinear two-wire lines, method of characteristics 

Zhuravlev V.M. Nonlinear models of wave processes in the dimension (1 + 1) and first order quasilinear equations. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2023;(3):143–158. (In Russ.). doi: 10.21685/2072-3040-2022-3-11