D'Alembert-type solution of the Cauchy problem for the Boussinesq-Klein-Gordon equation

In this paper we construct a weakly-nonlinear d'Alembert-type solution of the Cauchy problem for the Boussinesq-Klein-Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localised solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple-scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert-type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right- and left-propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have non-zero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localised solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have non-zero mean values. Importantly, in all cases the initial conditions for the leading-order Ostrovsky equations by construction have zero mean, while initial conditions for the Boussinesq-Klein-Gordon equation may have non-zero mean values.