Steady-state solutions of a dispersive system of nonlinear waves: PMM vol. 36, n≗5, 1972, pp. 811–821

Abstract

An “averaged Lagrangian” method of obtaining a dispersive system describing slow variations of the wave parameters for quasi-stationary waves, is obtained for equations admitting the existence of wave solutions. In the first “adiabatic” approximation the dispersive system for the Klein-Gordon equation represents a quasi-linear system of the hyperbolic type and admits discontinuous solutions. The structure of the discontinuities for the conservative and the nonconservative cases is investigated and the number of free parameters in a discontinuity determined. Various asymptotic methods find wide application in investigating nonlinear waves with dispersion [1, 2]. An adiabatic approximation method of obtaining a dispersive system of equations was proposed in [3] for the quasi-stationary waves, i. e. for the waves in which the change in the wave parameters is slow compared with the fundamental oscillations. That method is based on the process of averaging over the fast variable appearing in equations equivalent to the initial equations and written in divergent form. It was shown first for the conservative systems [4] and then for the nonconservative ones [5] that the dispersive system can be obtained from a variational equation averaged in the appropriate manner. Such averaging processes representing integration over a part of the independent variables find use in the theory of stress and strain in shells and rods when the Bubnov method is used, and also in the asymptotic theory of nonlinear oscillations [6].